Estimate limits from tables

Learning goals

Estimates limits from tables.
Compares function values as inputs approach the target from each side.
Distinguishes finite, infinite, one-sided, and nonexistent behavior when relevant.
Explains conclusions using limit notation.

Difficulty: 1 – 5

Click problems to add them to your worksheet.

Estimate limits from tables 1

The table gives values of $f(x)$ near $x=2$. <table><tr><th>$x$</th><td>$1.9$</td><td>$1.99$</td><td>$1.999$</td><td>$2.001$</td><td>$2.01$</td><td>$2.1$</td></tr><tr><th>$f(x)$</th><td>$4.81$</td><td>$4.981$</td><td>$4.9981$</td><td>$5.0019$</td><td>$5.019$</td><td>$5.19$</td></tr></table> Estimate $\lim_{x\to2}f(x)$.

Hints

- Compare the outputs for x-values closest to $2$. - Look at the left and right sides separately first. - A two-sided estimate is supported when both sides approach the same value.

Solution

1. For x-values less than $2$, the outputs approach $5$. 2. For x-values greater than $2$, the outputs also approach $5$. 3. Therefore $\lim_{x\to2}f(x)\approx 5$.

Answer

$\lim_{x\to2}f(x)\approx 5$.

Estimate limits from tables 2

The table shows the behavior of $G(x)$ for larger and larger positive inputs. <table><tr><th>$x$</th><td>$10$</td><td>$100$</td><td>$1000$</td><td>$10{,}000$</td></tr><tr><th>$G(x)$</th><td>$4$</td><td>$16$</td><td>$64$</td><td>$256$</td></tr></table> Does the table support a finite value for $\lim_{x\to\infty}G(x)$? Describe the trend instead.

Hints

- Look for whether the output row is leveling off. - Compare the size of successive outputs as the inputs grow. - Distinguish approaching a finite number from growing without bound.

Solution

1. As the inputs increase, the outputs grow from $4$ to $256$ rather than leveling off. 2. The displayed values provide no evidence of approaching a finite number. 3. The table suggests unbounded positive growth as $x\to\infty$, so no finite limit is supported.

Answer

No finite limit is supported. The table suggests $G(x)\to+\infty$ as $x\to\infty$.

Estimate limits from tables 3

The table shows values of $J(x)$ for increasingly negative inputs. <table><tr><th>$x$</th><td>$-10$</td><td>$-100$</td><td>$-1000$</td><td>$-10{,}000$</td></tr><tr><th>$J(x)$</th><td>$-0.25$</td><td>$-0.025$</td><td>$-0.0025$</td><td>$-0.00025$</td></tr></table> Estimate $\lim_{x\to-\infty}J(x)$. From which side of the limiting y-value do the table values approach?

Hints

- Compare the magnitudes of the outputs as the inputs decrease without bound. - Notice the sign of every output. - A limit of zero can be approached entirely from negative values.

Solution

1. As $x$ becomes more negative, the outputs become closer to $0$ in magnitude. 2. Every displayed output is negative, so the values approach $0$ from below. 3. Therefore $\lim_{x\to-\infty}J(x)\approx0$.

Answer

$\lim_{x\to-\infty}J(x)\approx0$, approached from below.

Estimate limits from tables 4

Angles are in radians. The table shows values of $\frac{\sin u}{u}$ close to $u=0$, rounded to five decimal places. <table><tr><th>$u$</th><td>$-0.2$</td><td>$-0.1$</td><td>$-0.01$</td><td>$0.01$</td><td>$0.1$</td><td>$0.2$</td></tr><tr><th>$\frac{\sin u}{u}$</th><td>0.99335</td><td>0.99833</td><td>0.99998</td><td>0.99998</td><td>0.99833</td><td>0.99335</td></tr></table> What does the table suggest for $\lim_{u\to0}\frac{\sin u}{u}$? Why is there no entry for $u=0$?

Hints

- Read the entries toward the middle of the table from both ends. - Compare the ratios on the negative and positive sides. - Can a quotient have a limit even when it cannot be evaluated at the target input?

Solution

1. From both sides, the ratios in the table get closer to $1$, suggesting a limit of $1$. 2. At $u=0$, the denominator is zero, so the ratio is undefined there. 3. The table suggests the limit; it does not prove it for all closer nonzero inputs.

Answer

The table suggests a limit of $1$. The ratio is undefined at $u=0$ because its denominator would be zero.

Estimate limits from tables 5

Estimate $\lim_{x\to2}f(x)$ from the table. Explain what the values on each side suggest. <table><tr><th>$x$</th><td>1.9</td><td>1.99</td><td>1.999</td><td>2.001</td><td>2.01</td><td>2.1</td></tr><tr><th>$f(x)$</th><td>2.8</td><td>2.98</td><td>2.998</td><td>3.002</td><td>3.02</td><td>3.2</td></tr></table>

Hints

- Identify which inputs are to the left and right of $2$. - Read the values closest to the target input. - Check whether both sets of outputs head toward one number.

Solution

1. As $x$ increases toward $2$, the left-hand values approach approximately $3$. 2. The right-hand values also get closer to $3$ as $x$ decreases toward $2$. 3. Both trends support an estimated two-sided limit of $3$. The finite data cannot prove it.

Answer

The table suggests $\lim_{x\to2}f(x)\approx3$; both sides approach $3$.

Estimate limits from tables 6

A table records values of $g$ near $x=0$. <table><tr><th>$x$</th><td>$-0.1$</td><td>$-0.01$</td><td>$0.01$</td><td>$0.1$</td></tr><tr><th>$g(x)$</th><td>$-1.9$</td><td>$-1.99$</td><td>1.99</td><td>1.9</td></tr></table> What two different values do the sides appear to approach? Does the table support one two-sided limit?

Hints

- Group the entries by the sign of the input. - Compare the outputs for inputs nearest zero. - A two-sided limit requires the two one-sided limits to agree.

Solution

1. From negative inputs, the values suggest a left-hand limit of $-2$. 2. From positive inputs, the values suggest a right-hand limit of $2$. 3. Because the trends disagree, the table does not support a single two-sided limit.

Answer

Left: $-2$; right: $2$. No, the sides do not appear to approach one value.

Estimate limits from tables 7

A savings account starts with $\$10{,}000$ and earns $6\%$ annual interest. The table gives its balance $B(n)$ after one year when interest is added $n$ times. <table><tr><th>Times interest is added, $n$</th><td>1</td><td>4</td><td>12</td><td>52</td><td>365</td><td>3650</td></tr><tr><th>Balance after one year (dollars)</th><td>10,600.00</td><td>10,613.64</td><td>10,616.78</td><td>10,618.00</td><td>10,618.31</td><td>10,618.36</td></tr></table> a) Estimate $\lim_{n\to\infty}B(n)$ to the nearest dollar using the table. b) How much more does the account earn with daily instead of monthly compounding? Use the table and round to the nearest cent. c) Can you determine the limiting balance to the nearest cent from these six rounded entries? Explain.

Hints

- Compare the last few balances as the number of interest payments increases. - Which two columns correspond to monthly and daily interest? - Compare the precision requested in part c) with the change between the last two balances.

Solution

1. The balances increase, but by smaller amounts as $n$ grows. The last values suggest a limit near $\$10{,}618$, to the nearest dollar. 2. The table gives a daily balance of $\$10{,}618.31$ and a monthly balance of $\$10{,}616.78$. Their difference is $\$1.53$. 3. No. The last two displayed balances are still $\$0.05$ apart, and the table contains only selected values. It does not determine the limiting balance to the nearest cent.

Answer

a) Approximately $\$10{,}618$. b) $\$1.53$ more. c) No. The displayed values still differ by $\$0.05$, so they do not fix the limiting balance to the nearest cent.

Estimate limits from tables 8

The table gives values of $a_n=\left(1+\frac{1}{n}\right)^n$, rounded to three decimal places. <table><tr><th>$n$</th><td>5</td><td>50</td><td>500</td><td>1000</td></tr><tr><th>$a_n$</th><td>2.488</td><td>2.692</td><td>2.716</td><td>2.717</td></tr></table> a) Use the last two entries to estimate $\lim_{n\to\infty}a_n$ to two decimal places. Explain how those entries support your estimate. b) Abena says, “The base approaches $1$, so the powers must approach $1$.” Explain why the table challenges her reasoning.

Hints

- Compare the entries for increasing $n$. - What changes besides the number inside the parentheses? - Use the final few entries as evidence for an estimate, not a proof.

Solution

1. The last two entries are $2.716$ and $2.717$; both round to $2.72$ to two decimal places. 2. This suggests a limit of $2.72$. The sequence is known to approach Euler's number, $e$, but a finite table does not prove this. 3. Both the base and the exponent change. Although the base gets closer to $1$, the exponent increases, and the values shown stay near $2.7$, not $1$.

Answer

a) About $2.72$, since both of the last two values round to $2.72$. This limit is Euler's number, $e$. b) The exponent grows while the base approaches $1$. The table suggests a limit near $2.72$, not $1$.

Estimate limits from tables 9

Use the table to determine whether $\lim_{x\to-1}g(x)$ exists. <table><tr><th>$x$</th><td>$-1.1$</td><td>$-1.01$</td><td>$-1.001$</td><td>$-0.999$</td><td>$-0.99$</td><td>$-0.9$</td></tr><tr><th>$g(x)$</th><td>$2.8$</td><td>$2.98$</td><td>$2.998$</td><td>$4.002$</td><td>$4.02$</td><td>$4.2$</td></tr></table>

Hints

- Split the table at $x=-1$. - Estimate the limiting output on each side. - Compare the two one-sided estimates before deciding on a two-sided limit.

Solution

1. From the left of $-1$, the values approach $3$. 2. From the right of $-1$, the values approach $4$. 3. Since the one-sided estimates disagree, $\lim_{x\to-1}g(x)$ does not exist.

Answer

The limit does not exist; the left-hand values approach $3$ and the right-hand values approach $4$.

Estimate limits from tables 10

A table samples $h(x)$ near $x=0$. <table><tr><th>$x$</th><td>$-0.1$</td><td>$-0.01$</td><td>$-0.001$</td><td>$0.001$</td><td>$0.01$</td><td>$0.1$</td></tr><tr><th>$h(x)$</th><td>$10$</td><td>$100$</td><td>$1000$</td><td>$1000$</td><td>$100$</td><td>$10$</td></tr></table> Does the table support a finite value for $\lim_{x\to0}h(x)$? Describe the observed behavior.

Hints

- Compare output size with distance from $0$. - Ask whether the outputs are stabilizing near a finite number. - Large growth on both sides signals unbounded behavior rather than a finite estimate.

Solution

1. On both sides of $0$, the outputs become much larger as $x$ gets closer to $0$. 2. The values do not settle near any finite number. 3. The table therefore does not support a finite limit; it suggests unbounded positive behavior near $0$.

Answer

No finite limit is supported. The table suggests unbounded positive behavior as $x\to0$ from both sides.

Estimate limits from tables 11

A table of values near $x=3$ is shown, and separately you are told that $q(3)=10$. <table><tr><th>$x$</th><td>$2.9$</td><td>$2.99$</td><td>$2.999$</td><td>$3.001$</td><td>$3.01$</td><td>$3.1$</td></tr><tr><th>$q(x)$</th><td>$6.8$</td><td>$6.98$</td><td>$6.998$</td><td>$7.002$</td><td>$7.02$</td><td>$7.2$</td></tr></table> Estimate $\lim_{x\to3}q(x)$. Explain the role of $q(3)=10$.

Hints

- Use the rows near but not equal to $x=3$. - Compare the values from the left and right. - Keep the function value at the point separate from the nearby trend.

Solution

1. The table values from both sides approach $7$. 2. Therefore $\lim_{x\to3}q(x)\approx7$. 3. The separate value $q(3)=10$ does not affect the estimate because the limit concerns nearby inputs rather than the single input $3$.

Answer

$\lim_{x\to3}q(x)\approx7$. The value $q(3)=10$ does not change the limit.

Estimate limits from tables 12

The table lists values of a function $m$ near $x=2.5$. The entries are not ordered by distance from $2.5$. <table><tr><th>$x$</th><td>$2.2$</td><td>$2.49$</td><td>$2.7$</td><td>$2.501$</td><td>$2.51$</td><td>$2.499$</td></tr><tr><th>$m(x)$</th><td>$3.10$</td><td>$3.245$</td><td>$3.35$</td><td>$3.2505$</td><td>$3.255$</td><td>$3.2495$</td></tr></table> Estimate $\lim_{x\to2.5}m(x)$, and identify which pair of table entries gives the strongest evidence for your estimate.

Hints

- First compare how close each input is to $2.5$. - Use evidence from both sides of the target input. - The most informative rows are usually those nearest the approach value.

Solution

1. The inputs closest to $2.5$ are $2.499$ and $2.501$, one on each side. 2. Their outputs are $3.2495$ and $3.2505$, which bracket $3.25$ very closely. 3. Therefore $\lim_{x\to2.5}m(x)\approx3.25$.

Answer

$\lim_{x\to2.5}m(x)\approx3.25$. The strongest evidence comes from $x=2.499$ and $x=2.501$.

Estimate limits from tables 13

A function $r$ is sampled near $x=-2$. <table><tr><th>$x$</th><td>$-2.2$</td><td>$-2.02$</td><td>$-2.002$</td><td>$-1.998$</td><td>$-1.98$</td><td>$-1.8$</td></tr><tr><th>$r(x)$</th><td>$-0.18$</td><td>$0.024$</td><td>$-0.003$</td><td>$0.004$</td><td>$-0.021$</td><td>$0.16$</td></tr></table> The outputs do not approach the limit monotonically. Estimate $\lim_{x\to-2}r(x)$ anyway, and explain what feature of the table matters more than monotonicity.

Hints

- Focus on distance from the target input rather than whether the outputs always increase or decrease. - Compare the magnitudes of the outputs nearest $-2$. - Check whether both sides are clustering around one value.

Solution

1. The outputs nearest $x=-2$ are $-0.003$ and $0.004$, both close to $0$. 2. Values on both sides become small in magnitude as the inputs move closer to $-2$, even though their signs vary. 3. Therefore $\lim_{x\to-2}r(x)\approx0$.

Answer

$\lim_{x\to-2}r(x)\approx0$. What matters is that values from both sides get close to the same number as $x$ gets close to $-2$; they do not need to move toward it monotonically.

Estimate limits from tables 14

Two functions are sampled near $x=1$. <table><tr><th>$x$</th><td>$0.9$</td><td>$0.99$</td><td>$0.999$</td><td>$1.001$</td><td>$1.01$</td><td>$1.1$</td></tr><tr><th>$a(x)$</th><td>$3.7$</td><td>$3.97$</td><td>$3.997$</td><td>$4.003$</td><td>$4.03$</td><td>$4.3$</td></tr><tr><th>$b(x)$</th><td>$3.8$</td><td>$3.98$</td><td>$3.998$</td><td>$5.002$</td><td>$5.02$</td><td>$5.2$</td></tr></table> Estimate $\lim_{x\to1}a(x)$ and determine whether $\lim_{x\to1}b(x)$ exists. Use the table to explain why the conclusions differ.

Hints

- Treat each function row separately. - For each function, compare the trend from inputs less than $1$ with the trend from inputs greater than $1$. - A two-sided limit requires the two one-sided trends to agree.

Solution

1. For $a$, values from the left approach $4$, and values from the right also approach $4$. Thus $\lim_{x\to1}a(x)\approx4$. 2. For $b$, left-side values approach $4$, while right-side values approach $5$. 3. Because the one-sided trends for $b$ disagree, $\lim_{x\to1}b(x)$ does not exist.

Answer

$\lim_{x\to1}a(x)\approx4$. $\lim_{x\to1}b(x)$ does not exist because the left-hand values approach $4$ while the right-hand values approach $5$.

Estimate limits from tables 15

The table gives values of $v(x)$ for increasingly large positive inputs. <table><tr><th>$x$</th><td>$10$</td><td>$100$</td><td>$1000$</td><td>$10{,}000$</td></tr><tr><th>$v(x)$</th><td>$1.7000$</td><td>$1.9700$</td><td>$1.9970$</td><td>$1.9997$</td></tr></table> Use the table to estimate $\lim_{x\to\infty}v(x)$. State one limitation of using only this finite table as evidence for an infinite-input limit.

Hints

- Read the output trend as the inputs become much larger. - Look for a value the outputs are getting closer to. - Distinguish numerical evidence from a proof about all large inputs.

Solution

1. As the inputs increase by powers of $10$, the outputs move closer to $2$: $1.7000, 1.9700, 1.9970, 1.9997$. 2. The table therefore supports the estimate $\lim_{x\to\infty}v(x)\approx2$. 3. A finite table cannot by itself prove what happens for all sufficiently large inputs; it only provides numerical evidence for the trend.

Answer

$\lim_{x\to\infty}v(x)\approx2$. The table supports the estimate but does not prove the behavior for arbitrarily large $x$.

Estimate limits from tables 16

A table contains only values with $x<6$. <table><tr><th>$x$</th><td>$5$</td><td>$5.8$</td><td>$5.98$</td><td>$5.998$</td></tr><tr><th>$h(x)$</th><td>$8.4$</td><td>$8.88$</td><td>$8.988$</td><td>$8.9988$</td></tr></table> What limit can reasonably be estimated from this table? Can the table alone determine $\lim_{x\to6}h(x)$? Explain.

Hints

- Check whether every listed input lies on the same side of $6$. - Identify which one-sided limit the data address. - Ask what additional side would be needed for a two-sided conclusion.

Solution

1. All listed inputs approach $6$ from the left, and the outputs approach $9$. 2. Thus the table supports $\lim_{x\to6^-}h(x)\approx9$. 3. The table provides no values for $x>6$, so it gives no evidence about the right-hand limit. Therefore it cannot by itself determine the two-sided limit.

Answer

The table supports $\lim_{x\to6^-}h(x)\approx9$. It does not by itself determine $\lim_{x\to6}h(x)$ because no right-side data are given.

Estimate limits from tables 17

Two tables sample the same function $F$ near $x=4$. <table><tr><th colspan="3">Coarser sample</th></tr><tr><th>$x$</th><td>$3.5$</td><td>$4.5$</td></tr><tr><th>$F(x)$</th><td>$9.4$</td><td>$10.6$</td></tr></table> <table><tr><th colspan="5">Finer sample</th></tr><tr><th>$x$</th><td>$3.99$</td><td>$3.999$</td><td>$4.001$</td><td>$4.01$</td></tr><tr><th>$F(x)$</th><td>$9.988$</td><td>$9.9988$</td><td>$10.0012$</td><td>$10.012$</td></tr></table> Estimate $\lim_{x\to4}F(x)$. Which table gives the more reliable estimate, and why?

Hints

- Compare how close each table's inputs are to $4$. - Look for a common value suggested by both sides. - Finer sampling near the target usually gives better evidence about a limit.

Solution

1. The coarse table suggests a value near $10$, since $9.4$ and $10.6$ lie on opposite sides of it. 2. The finer table gives outputs extremely close to $10$ for inputs much closer to $4$ from both sides. 3. Thus $\lim_{x\to4}F(x)\approx10$, with the finer table providing stronger numerical evidence.

Answer

$\lim_{x\to4}F(x)\approx10$. The finer table is more reliable because its inputs lie much closer to $4$ on both sides.

Estimate limits from tables 18

The table samples a function $Q$ only for inputs less than $-4$. <table><tr><th>$x$</th><td>$-5$</td><td>$-4.1$</td><td>$-4.01$</td><td>$-4.001$</td></tr><tr><th>$Q(x)$</th><td>$8$</td><td>$80$</td><td>$800$</td><td>$8000$</td></tr></table> Describe the behavior suggested by the table as $x\to-4^-$. Does the table support a finite left-hand limit?

Hints

- Track the output size as the inputs move closer to $-4$. - Ask whether the outputs are stabilizing near a finite number. - Keep the one-sided direction in view when stating the behavior.

Solution

1. As the inputs approach $-4$ from the left, the outputs increase from $8$ to $8000$. 2. The values grow rather than settling near a finite number. 3. The table suggests $Q(x)\to+\infty$ as $x\to-4^-$, so it does not support a finite left-hand limit.

Answer

The table suggests $Q(x)\to+\infty$ as $x\to-4^-$. No finite left-hand limit is supported.

Estimate limits from tables 19

The table samples $f(x)$ closer and closer to $x=0$. <table><tr><th>$x$</th><td>$-0.1$</td><td>$-0.01$</td><td>$-0.001$</td><td>$0.001$</td><td>$0.01$</td><td>$0.1$</td></tr><tr><th>$f(x)$</th><td>$1$</td><td>$-1$</td><td>$1$</td><td>$-1$</td><td>$1$</td><td>$-1$</td></tr></table> Does this table support a single value for $\lim_{x\to0}f(x)$? Explain what feature of the data prevents a limit estimate.

Hints

- Ignore whether the inputs are positive or negative at first and focus on the output pattern near $0$. - Ask whether the outputs settle into a smaller neighborhood around one number. - Persistent oscillation is different from ordinary numerical noise that shrinks near the target.

Solution

1. The displayed inputs get progressively closer to $0$, but the outputs continue to alternate between $1$ and $-1$. 2. The values do not cluster around one number as the inputs approach $0$. 3. Therefore the table does not support a single two-sided limit value.

Answer

No. The outputs keep switching between $1$ and $-1$ instead of approaching one value as $x\to0$.

Estimate limits from tables 20

The table samples a function $A$ at larger and larger positive inputs. <table><tr><th>$x$</th><td>$20$</td><td>$200$</td><td>$2000$</td><td>$20{,}000$</td></tr><tr><th>$A(x)$</th><td>$5.20$</td><td>$4.95$</td><td>$5.012$</td><td>$4.997$</td></tr></table> Estimate $\lim_{x\to\infty}A(x)$. Why does alternating above and below the estimated limit not by itself prevent convergence?

Hints

- Compare each output with $5$. - Focus on the size of the deviation, not only its sign. - A convergent sequence of sampled values need not approach from only one side.

Solution

1. The outputs alternate around $5$: first above, then below, then above, then below. 2. Their deviations from $5$ become small as the inputs grow. 3. The table therefore supports $\lim_{x\to\infty}A(x)\approx5$; crossing the limiting value is compatible with convergence when the values get progressively closer to it.

Answer

$\lim_{x\to\infty}A(x)\approx5$. Alternating sides does not prevent convergence when the outputs become closer and closer to the same value.

Estimate limits from tables 21

The x-values in this table are not equally spaced around $10$. <table><tr><th>$x$</th><td>$9.7$</td><td>$9.97$</td><td>$9.997$</td><td>$10.0005$</td><td>$10.02$</td><td>$10.4$</td></tr><tr><th>$C(x)$</th><td>$1.85$</td><td>$1.985$</td><td>$1.9985$</td><td>$2.00025$</td><td>$2.01$</td><td>$2.20$</td></tr></table> Estimate $\lim_{x\to10}C(x)$. Explain why equal spacing on the two sides of $10$ is not required for a numerical limit estimate.

Hints

- Identify the nearest inputs on each side of $10$. - Compare their outputs rather than trying to pair equal distances. - A limit concerns approach from both sides, not symmetry of the table.

Solution

1. Inputs on both sides of $10$ get very close to $10$. 2. The corresponding nearby outputs, such as $1.9985$ and $2.00025$, are close to $2$. 3. Thus $\lim_{x\to10}C(x)\approx2$. Equal spacing is unnecessary because the key question is whether values from each side approach the same output.

Answer

$\lim_{x\to10}C(x)\approx2$. The two sides need not use matching distances from $10$; they only need to provide values increasingly close to the target from both directions.

Estimate limits from tables 22

A single data set samples the same function near two different inputs. <table><tr><th>$x$</th><td>$-1.01$</td><td>$-1.001$</td><td>$-0.999$</td><td>$-0.99$</td><td>$1.99$</td><td>$1.999$</td><td>$2.001$</td><td>$2.01$</td></tr><tr><th>$E(x)$</th><td>$3.96$</td><td>$3.996$</td><td>$4.004$</td><td>$4.04$</td><td>$-2.97$</td><td>$-2.997$</td><td>$-3.003$</td><td>$-3.03$</td></tr></table> Estimate $\lim_{x\to-1}E(x)$ and $\lim_{x\to2}E(x)$. Explain why values near the other target should be ignored for each estimate.

Hints

- Separate the table into the cluster near $-1$ and the cluster near $2$. - For each target, use values from both sides of that target only. - Do not mix data from neighborhoods centered at different inputs.

Solution

1. Near $x=-1$, values from both sides approach $4$, so $\lim_{x\to-1}E(x)\approx4$. 2. Near $x=2$, values from both sides approach $-3$, so $\lim_{x\to2}E(x)\approx-3$. 3. A limit at one target depends on behavior near that target; rows clustered around the other input are not relevant evidence.

Answer

$\lim_{x\to-1}E(x)\approx4$ and $\lim_{x\to2}E(x)\approx-3$. For each estimate, only values near that approach point are relevant; values clustered near the other target should be ignored.

Estimate limits from tables 23

Only a coarse numerical sample is available near $x=-5$. <table><tr><th>$x$</th><td>$-5.4$</td><td>$-5.2$</td><td>$-4.8$</td><td>$-4.6$</td></tr><tr><th>$F(x)$</th><td>$5.6$</td><td>$5.8$</td><td>$6.2$</td><td>$6.4$</td></tr></table> Which estimate is best supported for $\lim_{x\to-5}F(x)$: $5$, $6$, or $7$? Explain why the estimate should be viewed as less precise than one based on inputs much closer to $-5$.

Hints

- Compare the values on the two sides of $-5$. - Look for the choice around which the outputs are centered. - Consider how close the nearest input actually is to the target.

Solution

1. The values below $-5$ are below $6$, while the values above $-5$ are above $6$. 2. The pattern is centered near $6$, so $6$ is the best-supported estimate among the choices. 3. Because the closest inputs are still $0.2$ away from $-5$, the table gives only coarse evidence and does not justify a highly precise estimate.

Answer

$6$ is the best-supported estimate. The data are relatively far from $-5$, so the table supports only a coarse approximation.

Estimate limits from tables 24

The table gives very small output values near $x=4$. <table><tr><th>$x$</th><td>$3.9$</td><td>$3.99$</td><td>$3.999$</td><td>$4.001$</td><td>$4.01$</td><td>$4.1$</td></tr><tr><th>$H(x)$</th><td>$0.0018$</td><td>$0.00198$</td><td>$0.001998$</td><td>$0.002002$</td><td>$0.00202$</td><td>$0.0022$</td></tr></table> Estimate $\lim_{x\to4}H(x)$ to four decimal places. Why would reporting the answer as $0$ lose important information?

Hints

- Focus on the entries paired with inputs closest to $4$. - Keep enough decimal places to distinguish a small nonzero value from zero. - Use both sides before applying the requested rounding.

Solution

1. The values nearest $x=4$ are $0.001998$ and $0.002002$. 2. These values approach $0.002$ from opposite sides. 3. To four decimal places, $\lim_{x\to4}H(x)\approx0.0020$. Reporting $0$ would discard the nonzero limiting value indicated by the data.

Answer

$\lim_{x\to4}H(x)\approx0.0020$. Writing $0$ would erase the small but clearly nonzero limit suggested by the table.

Estimate limits from tables 25

A function $K$ is defined for $x>0$. The table samples values near the boundary $x=0$. <table><tr><th>$x$</th><td>$0.5$</td><td>$0.1$</td><td>$0.01$</td><td>$0.001$</td></tr><tr><th>$K(x)$</th><td>$1.581$</td><td>$1.449$</td><td>$1.418$</td><td>$1.415$</td></tr></table> Estimate $\lim_{x\to0^+}K(x)$ to three decimal places. Why is a right-hand limit the natural question here?

Hints

- Use the stated domain before interpreting the table. - Follow the outputs as the positive inputs approach $0$. - Match the limit direction to the side of the domain that is available.

Solution

1. The function is defined only for positive inputs, so values can approach $0$ only from the right within the domain. 2. As the positive inputs get closer to $0$, the outputs approach about $1.414$. 3. Therefore $\lim_{x\to0^+}K(x)\approx1.414$.

Answer

$\lim_{x\to0^+}K(x)\approx1.414$. A right-hand limit is natural because the domain contains nearby inputs only for $x>0$.

Estimate limits from tables 26

A calculator displays the following rounded values of $L(x)$ near $x=8$. <table><tr><th>$x$</th><td>$7.9$</td><td>$7.99$</td><td>$7.999$</td><td>$8.001$</td><td>$8.01$</td><td>$8.1$</td></tr><tr><th>$L(x)$</th><td>$12.0$</td><td>$12.0$</td><td>$12.0$</td><td>$12.0$</td><td>$12.0$</td><td>$12.0$</td></tr></table> What limit estimate is supported by the displayed table? Explain why the rounded display does not prove that every nearby value of $L(x)$ is exactly $12$.

Hints

- Separate what the displayed values support from what they prove exactly. - Notice the stated rounding of the calculator output. - A numerical table can hide small differences that occur beyond the displayed digits.

Solution

1. All displayed values on both sides of $8$ are $12.0$, so the table supports $\lim_{x\to8}L(x)\approx12$. 2. Each output has been rounded to one decimal place. 3. Different nearby values can round to the same display, so the table does not prove that $L(x)=12$ exactly for every nearby input.

Answer

The table supports $\lim_{x\to8}L(x)\approx12$. Because the outputs are rounded, identical displays do not imply the exact nearby function values are all $12$.

Estimate limits from tables 27

Maya has these function values near $x=4$: <table><tr><th>$x$</th><td>4.1</td><td>4.01</td><td>4.001</td></tr><tr><th>$f(x)$</th><td>8.2</td><td>8.02</td><td>8.002</td></tr></table> She can add two entries from this list: <table><tr><th>$x$</th><td>3.5</td><td>3.9</td><td>3.99</td><td>3.999</td></tr><tr><th>$f(x)$</th><td>7.0</td><td>7.8</td><td>7.98</td><td>7.998</td></tr></table> a) Which two entries would best check the limit from the left? b) What two-sided limit does the completed table suggest? Could these five values prove it? Explain.

Hints

- On which side of $4$ are the original measurements? - How much would a sample far from the target help you estimate the nearby behavior? - Do the outputs from both sides appear to settle near the same height? What can a finite table establish?

Solution

1. The original table gives values only for $x>4$. For the left-hand approach, the closest available inputs are $x=3.99$ and $x=3.999$, with outputs $7.98$ and $7.998$. 2. As inputs approach $4$ from the left, the outputs suggest $8$ from below. The values on the right suggest $8$ from above. 3. Thus the table suggests $\lim_{x\to4}f(x)=8$, but a finite set of values cannot prove how the function behaves at every closer input.

Answer

a) Use $(3.99,7.98)$ and $(3.999,7.998)$. b) The estimated limit is $8$. No finite table can prove the limit.

Estimate limits from tables 28

A table lists $f(x)=2$ at $x=-0.1,-0.01,0.01,0.1$. A classmate says, “That proves $\lim_{x\to0}f(x)=2$.” Describe a different function that also has value $2$ at all four listed inputs but does not have limit $2$ as $x\to0$.

Hints

- Do four specified function values determine the entire function? - What does the table tell you about unlisted inputs closer to zero? - Compare what a finite table can establish with what a limit statement claims.

Solution

1. Define $g(x)=2$ at the four listed inputs and $g(x)=4$ at every other nonzero input near $0$. 2. This function matches every entry in the given table. 3. Since all sufficiently close nonzero inputs other than those four have output $4$, $\lim_{x\to0}g(x)=4$, not $2$. 4. Thus a finite table cannot prove the proposed limit.

Answer

One example is $g(x)=2$ at the four sample inputs and $g(x)=4$ at all other inputs near $0$. Its limit at $0$ is $4$, even though the table matches.

Estimate limits from tables 29

Agustín wants to estimate $\lim_{x\to3}p(x)$ from this table. <table><tr><th>$x$</th><td>$2.9$</td><td>$2.99$</td><td>$2.999$</td><td>$3.001$</td><td>$3.01$</td><td>$3.1$</td></tr><tr><th>$p(x)$</th><td>$7.2$</td><td>$7.92$</td><td>$7.992$</td><td>$8.008$</td><td>$8.08$</td><td>$8.8$</td></tr></table> Agustín averages the two farthest outputs, $7.2$ and $8.8$, and gets $8$. The numerical answer happens to be reasonable. Explain why this method is not a reliable general method for estimating a limit, and give the estimate supported by the table.

Hints

- Ask which rows best represent behavior near $x=3$. - A correct numerical result can come from an unjustified method. - Compare the closest outputs on opposite sides of the target.

Solution

1. A limit is determined by behavior for inputs close to the target, so the farthest rows are weaker evidence than the nearest rows. 2. The closest outputs, $7.992$ and $8.008$, approach $8$ from opposite sides. 3. Thus the table supports $\lim_{x\to3}p(x)\approx8$, but not because the farthest outputs happen to average to $8$.

Answer

$\lim_{x\to3}p(x)\approx8$. Averaging faraway outputs is not a reliable limit procedure; the estimate is supported by the values at inputs closest to $3$.

Estimate limits from tables 30

The graph and table show the same function $h$ near $x=1$. The graph is displayed at a coarse vertical scale. <table><tr><th>$x$</th><td>$0.9$</td><td>$0.99$</td><td>$0.999$</td><td>$1.001$</td><td>$1.01$</td><td>$1.1$</td></tr><tr><th>$h(x)$</th><td>$2.00100$</td><td>$2.00370$</td><td>$2.00397$</td><td>$1.99603$</td><td>$1.99630$</td><td>$1.99900$</td></tr></table> a) Using only the graph and reporting to the nearest hundredth, estimate the left-hand and right-hand limits at $x=1$. b) Using the table, estimate each one-sided limit to the nearest thousandth. c) Does $\lim_{x\to1}h(x)$ exist? Explain why the coarse graphical estimates in part a) do not settle the question.

Hints

- Keep the precision requested in each part; the graph and table do not resolve values equally finely. - For the table, separate inputs less than $1$ from inputs greater than $1$ and focus on the closest entries. - A two-sided limit requires the actual one-sided limits to agree, not merely rounded estimates that look the same.

Solution

1. At the graph's displayed precision, both branches appear to approach about $2.00$, so the nearest-hundredth graphical estimates agree. 2. The left-side values closest to $1$ approach $2.004$. 3. The right-side values closest to $1$ approach $1.996$. 4. Because $2.004\ne1.996$, the two-sided limit does not exist. 5. The actual separation is only $0.008$, so rounding graphical estimates to the nearest hundredth hides it. The table supplies the extra precision needed to distinguish the one-sided limits.

Answer

a) Left-hand limit $\approx2.00$; right-hand limit $\approx2.00$. b) Left-hand limit $\approx2.004$; right-hand limit $\approx1.996$. c) No. $\lim_{x\to1}h(x)$ does not exist because the one-sided limits are unequal. The graph's nearest-hundredth precision rounds both to $2.00$, masking the difference.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Calculus math worksheets

Estimate limits from tables

Learning goals

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution