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Estimate limits from graphs

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54266812
Use the graph to estimate \(\lim_{x\to2.5}p(x)\) to the nearest \(0.25\).
Figure for problem 542668

Hints

- Locate \(x=2.5\) halfway between the labeled integer inputs. - Read the graph height using the quarter-unit horizontal grid. - Check that the same height is approached from both sides.

Solution

1. Trace the graph from both sides toward \(x=2.5\). 2. The y-values approach approximately \(2.75\) from both sides. 3. Therefore \(\lim_{x\to2.5}p(x)\approx2.75\).

Answer

\(\lim_{x\to2.5}p(x)\approx2.75\).
54248112
Use the graph of \(f\) to estimate \(\lim_{x\to1}f(x)\). Compare this limit with \(f(1)\), and explain whether the value at \(x=1\) affects the limit.
Figure for problem 542481

Hints

- Follow the curve toward \(x=1\) from both sides. - Use the y-value approached by the curve, not automatically the filled point. - Check whether the left and right sides approach the same height.

Solution

1. From both sides of \(x=1\), the graph approaches the open point \((1, 0)\). 2. Thus the left-hand and right-hand limits both equal \(0\). 3. Therefore \(\lim_{x\to1}f(x)=0\). 4. The filled point at \((1, 3)\) gives \(f(1)=3\) but does not change the nearby limiting behavior.

Answer

\(\lim_{x\to1}f(x)=0\). The filled point gives \(f(1)=3\), which does not determine the limit.
54248212
Use the graph of \(g\) to estimate \(\lim_{x\to-1^-}g(x)\) and \(\lim_{x\to-1^+}g(x)\). Then determine whether \(\lim_{x\to-1}g(x)\) exists.
Figure for problem 542482

Hints

- Trace the graph toward \(x=-1\) separately from the left and from the right. - Read the height approached on each side. - A two-sided limit exists only when the two one-sided values agree.

Solution

1. From the left, the graph approaches the open point at \((-1, 2)\), so \(\lim_{x\to-1^-}g(x)=2\). 2. From the right, the graph approaches the open point at \((-1, -1)\), so \(\lim_{x\to-1^+}g(x)=-1\). 3. Because the one-sided limits are unequal, \(\lim_{x\to-1}g(x)\) does not exist.

Answer

\(\lim_{x\to-1^-}g(x)=2\), \(\lim_{x\to-1^+}g(x)=-1\), and \(\lim_{x\to-1}g(x)\) does not exist.
54248412
Use the graph of \(g\) to estimate \(\lim_{x\to1}g(x)\). Describe the local shape at \(x=1\), and decide whether that feature prevents the limit from existing.
Figure for problem 542484

Hints

- A limit depends on the y-value approached, not on whether the graph is smooth. - Trace both arms of the graph toward \(x=1\). - Compare the two one-sided heights.

Solution

1. From the left, the graph approaches the point \((1, 2)\). 2. From the right, the graph also approaches \((1, 2)\). 3. Thus \(\lim_{x\to1}g(x)=2\). 4. A corner does not prevent a limit when both sides approach the same y-value.

Answer

\(\lim_{x\to1}g(x)=2\). The corner does not prevent the limit from existing.
54248512
Use the graph of \(r\) to describe the behavior of \(r(x)\) as \(x\to0^-\) and as \(x\to0^+\). Does \(\lim_{x\to0}r(x)\) exist as a finite real number?
Figure for problem 542485

Hints

- Watch what happens to the height of each branch near \(x=0\). - Ask whether the branches settle near any finite y-value. - A graph leaving every finite vertical scale indicates unbounded behavior rather than a finite limit.

Solution

1. On both sides of \(x=0\), the graph rises beyond the visible y-range as \(x\) moves closer to \(0\). 2. The y-values do not approach any finite height. 3. Therefore \(\lim_{x\to0}r(x)\) does not exist as a finite real number; the behavior is unbounded above.

Answer

No finite real limit exists. The graph is unbounded above as \(x\to0\) from both sides.
54248812
Use the graph to estimate \(\lim_{x\to1.5}p(x)\). The grid marks every \(0.5\) unit on both axes.
Figure for problem 542488

Hints

- Locate \(x=1.5\) using the half-unit grid. - Follow the curve toward that vertical line from both sides. - Read the corresponding y-value using the same grid spacing.

Solution

1. The curve is continuous near \(x=1.5\). 2. Reading the graph at \(x=1.5\), the y-value approached from both sides is about \(1.0\). 3. Therefore \(\lim_{x\to1.5}p(x)\approx1.0\).

Answer

\(\lim_{x\to1.5}p(x)\approx 1.0\).
54266312
Use the graph of \(f\) to describe \(f(x)\) as \(x\to0^-\) and as \(x\to0^+\). Does a finite two-sided limit exist?
Figure for problem 542663

Hints

- Read each branch separately as it moves toward the y-axis. - Distinguish decreasing without bound from increasing without bound. - Compare the two one-sided behaviors before stating a two-sided conclusion.

Solution

1. From the left, the graph decreases without bound as it approaches \(x=0\), so \(f(x)\to-\infty\). 2. From the right, the graph increases without bound as it approaches \(x=0\), so \(f(x)\to+\infty\). 3. The one-sided behaviors do not approach the same finite value, so no finite two-sided limit exists.

Answer

\(\lim_{x\to0^-}f(x)=-\infty\), \(\lim_{x\to0^+}f(x)=+\infty\), and no finite two-sided limit exists.
54266912
The graph shows a function \(q\) on its full real domain. Estimate \(\lim_{x\to-3^+}q(x)\) and \(\lim_{x\to3^-}q(x)\). Explain why the opposite one-sided limits are not real-domain questions.
Figure for problem 542669

Hints

- Identify the two endpoints of the displayed domain. - Approach each endpoint from the side where the graph exists. - Use the fact that the graph shows the full real domain when deciding whether an opposite-side approach is available.

Solution

1. As \(x\to-3^+\), the graph approaches y-value \(0\), so the right-hand limit is \(0\). 2. As \(x\to3^-\), the graph also approaches y-value \(0\), so the left-hand limit is \(0\). 3. The displayed full real domain has no inputs below \(-3\) or above \(3\), so the opposite one-sided approaches are unavailable in the real domain.

Answer

\(\lim_{x\to-3^+}q(x)=0\) and \(\lim_{x\to3^-}q(x)=0\). The opposite approaches are outside the function's real domain.
54267412
The graph shows a function \(h\) on its full real domain. Estimate \(\lim_{x\to0^+}h(x)\). Why is a left-hand estimate from the graph not part of this real-valued problem?
Figure for problem 542674

Hints

- Follow the graph toward its left endpoint. - Read the y-value approached as \(x\) gets close to \(0\) from within the domain. - Use the fact that the graph displays the full real domain.

Solution

1. As \(x\) approaches \(0\) through positive values, the graph approaches \((0,0)\). 2. Therefore \(\lim_{x\to0^+}h(x)=0\). 3. The displayed full real domain contains no inputs with \(x<0\), so there is no real graph to approach \(0\) from the left.

Answer

\(\lim_{x\to0^+}h(x)=0\). There is no real-valued left-hand approach because the displayed domain has no inputs with \(x<0\).
55592012
Use the graph of \(f\) at \(x=1\). Write, in limit notation, the left-hand limit and the right-hand limit. State \(f(1)\), and determine whether the two-sided limit exists.
Figure for problem 555920

Hints

- Read the height approached by each branch separately. - Use the open points for one-sided limiting behavior and the filled point for the function value. - Compare the two one-sided limits before deciding whether a two-sided limit exists.

Solution

1. Following the left branch toward \(x=1\), the outputs approach \(2\). Thus \(\lim_{x\to1^-}f(x)=2\). 2. Following the right branch toward \(x=1\), the outputs approach \(4\). Thus \(\lim_{x\to1^+}f(x)=4\). 3. The filled point is at \((1,3)\), so \(f(1)=3\). 4. Because the one-sided limits are unequal, \(\lim_{x\to1}f(x)\) does not exist.

Answer

\(\lim_{x\to1^-}f(x)=2\) \(\lim_{x\to1^+}f(x)=4\) \(f(1)=3\) \(\lim_{x\to1}f(x)\) does not exist.
53498612
Use the graph of the piecewise-defined function \(f\) to determine whether \(\lim_{x\to1}f(x)\) and \(\lim_{x\to3}f(x)\) exist. Justify each answer by describing the behavior from the left and from the right.
Figure for problem 534986

Hints

- Trace the graph toward each input from the left and record the y-value approached. - Repeat from the right. - A two-sided limit exists only when the two one-sided limits agree. - The function value at the input does not determine the limit by itself.

Solution

1. As \(x\) approaches \(1\) from the left, \(f(x)\) approaches \(2\). As \(x\) approaches \(1\) from the right, \(f(x)\) also approaches \(2\). Therefore, \(\lim_{x\to1}f(x)=2\). The separate value \(f(1)=1\) does not affect the existence or value of the limit. 2. As \(x\) approaches \(3\) from the left, \(f(x)\) approaches \(3\). As \(x\) approaches \(3\) from the right, \(f(x)\) approaches \(1\). Since the one-sided limits are different, \(\lim_{x\to3}f(x)\) does not exist.

Answer

At \(x=1\): \(\lim_{x\to1}f(x)=2\). At \(x=3\): \(\lim_{x\to3}f(x)\) does not exist because the left-hand limit is \(3\) and the right-hand limit is \(1\).
54248312
The graph shows a function \(p\) near \(x=0\). Determine whether \(\lim_{x\to0}p(x)\) exists as a finite number. Justify your conclusion from the oscillatory behavior near \(x=0\).
Figure for problem 542483

Hints

- Compare the vertical spread of the oscillations as the graph approaches the y-axis. - Ask whether the oscillation height narrows toward one y-value. - A finite limit requires all sufficiently nearby outputs to stay close to one number.

Solution

1. On both sides of \(x=0\), the graph keeps oscillating between separated y-values as it gets closer to the y-axis. 2. The vertical spread does not narrow toward one height, so the nearby outputs do not settle near a single number. 3. Therefore \(\lim_{x\to0}p(x)\) does not exist.

Answer

No. \(\lim_{x\to0}p(x)\) does not exist because the oscillations continue with nonshrinking amplitude instead of approaching one value.
54248612
Use the graph of \(r\) to answer the questions. a) Estimate \(\lim_{x\to0^-}r(x)\). b) Describe the behavior of \(r(x)\) as \(x\to0^+\). c) Does a finite two-sided limit exist? Justify your conclusion from the two branches.
Figure for problem 542486

Hints

- Analyze the two sides separately before combining them. - On the right, ask whether the branch stays near any finite height. - A finite two-sided limit requires the same finite value from both sides.

Solution

1. The left branch is horizontal at y-value \(2\), so \(\lim_{x\to0^-}r(x)=2\). 2. The right branch rises without bound as \(x\to0^+\). 3. Since the two sides do not approach the same finite value, no finite two-sided limit exists.

Answer

a) \(2\) b) \(r(x)\) increases without bound as \(x\to0^+\). c) No finite two-sided limit exists.
54248712
Three panels show different behaviors near \(x=0\). a) Does \(\lim_{x\to0}f(x)\) exist? If so, estimate it. b) Does \(\lim_{x\to0}g(x)\) exist? c) Does \(\lim_{x\to0}h(x)\) exist? Use the graphical behavior in each panel to justify your answers.
Figure for problem 542487

Hints

- Analyze each panel independently. - For panels a) and b), compare the y-values approached from the left and from the right. - In panel c), look at whether the oscillations shrink toward one height or continue to span a nonzero vertical range.

Solution

1. In panel a), both branches approach \(2\), so \(\lim_{x\to0}f(x)=2\). 2. In panel b), the left branch approaches \(-1\) and the right branch approaches \(3\). Since the one-sided limits are different, the two-sided limit does not exist. 3. In panel c), the graph continues to oscillate between values near \(1\) and \(2\) as \(x\) approaches \(0\). The outputs do not settle near one number, so the two-sided limit does not exist.

Answer

a) \(\lim_{x\to0}f(x)=2\). b) \(\lim_{x\to0}g(x)\) does not exist. c) \(\lim_{x\to0}h(x)\) does not exist.
54248912
Use the graph of \(F\) to analyze three x-values. a) Estimate \(\lim_{x\to-2}F(x)\). b) Determine whether \(\lim_{x\to0}F(x)\) exists. c) Estimate \(\lim_{x\to2}F(x)\). Justify each conclusion from the left- and right-side behavior.
Figure for problem 542489

Hints

- Treat each x-value as a separate local question. - At \(-2\), distinguish the open point from the filled point. - At \(0\), compare the two branch heights. - At \(2\), follow the same continuous branch from both sides.

Solution

1. At \(x=-2\), both sides of the first branch approach y-value \(1\), so \(\lim_{x\to-2}F(x)=1\), even though the plotted function value is different. 2. At \(x=0\), the left branch approaches \(2\) while the right branch approaches \(-2\), so the two-sided limit does not exist. 3. At \(x=2\), the right-side branch is continuous through y-value \(-1\), and nearby values from both sides of \(2\) on that branch approach \(-1\). 4. Thus \(\lim_{x\to2}F(x)=-1\).

Answer

a) \(1\) b) \(\lim_{x\to0}F(x)\) does not exist. c) \(-1\)
54266412
The graph shows \(g\) near \(x=0\). Use the graph to estimate \(\lim_{x\to0}g(x)\). Explain how the changing height of the oscillations affects your conclusion.
Figure for problem 542664

Hints

- Track the vertical size of the oscillations as the graph approaches \(x=0\). - Compare the heights of successive peaks and troughs closer to the y-axis. - A graph can oscillate and still suggest a limit if the entire oscillation narrows toward one y-value.

Solution

1. The graph oscillates above and below the x-axis on both sides of \(x=0\). 2. As the graph gets closer to \(x=0\), both the peaks and the troughs move closer to y-value \(0\), so the vertical spread of the oscillation shrinks. 3. The nearby outputs are therefore confined closer and closer to \(0\) from both sides. 4. Thus the graph supports \(\lim_{x\to0}g(x)=0\).

Answer

\(\lim_{x\to0}g(x)=0\). The oscillations continue, but their amplitude shrinks to \(0\).
54266512
Use the graph of \(h\) to estimate \(\lim_{x\to-1^-}h(x)\), \(\lim_{x\to-1^+}h(x)\), and \(\lim_{x\to\infty}h(x)\).
Figure for problem 542665

Hints

- Inspect the two branches next to \(x=-1\) separately. - Use the direction of unbounded motion to choose the sign of infinity. - For the end behavior, look for the y-value the graph approaches as \(x\) moves right.

Solution

1. The left branch decreases without bound as \(x\to-1^-\), so the left-hand limit is \(-\infty\). 2. The right branch increases without bound as \(x\to-1^+\), so the right-hand limit is \(+\infty\). 3. Far to the right, the graph levels off near y-value \(1\), so \(\lim_{x\to\infty}h(x)=1\).

Answer

\(\lim_{x\to-1^-}h(x)=-\infty\), \(\lim_{x\to-1^+}h(x)=+\infty\), and \(\lim_{x\to\infty}h(x)=1\).
54266612
The graph shows \(f\). Use it to determine \(\lim_{x\to0}|f(x)|\). Explain whether the filled point at \(x=0\) affects this limit.
Figure for problem 542666

Hints

- First read the limit of the original graph near \(x=0\). - Then apply the absolute value to the y-value being approached. - Separate the nearby behavior from the isolated filled point.

Solution

1. Both graph branches approach y-value \(-2\) as \(x\to0\), so \(f(x)\to-2\). 2. Taking absolute values makes the nearby outputs approach \(|-2|=2\). 3. The filled point gives \(f(0)=3\), but a single point value does not change the limit.

Answer

\(\lim_{x\to0}|f(x)|=2\). The filled point does not affect the limit.
54266712
Use the graph to analyze the inputs \(x=-2\), \(x=1\), and \(x=3\). At which of these inputs does a finite two-sided limit exist, and what is its value?
Figure for problem 542667

Hints

- For each input, compare the y-values approached from the left and right. - An isolated filled point does not determine a limit. - Unbounded behavior is not a finite limit.

Solution

1. At \(x=-2\), both branches approach y-value \(1\), so the finite two-sided limit is \(1\). 2. At \(x=1\), the left branch approaches \(2\) and the right branch approaches \(-1\), so the two-sided limit does not exist. 3. At \(x=3\), the graph is unbounded on the two sides, so no finite two-sided limit exists. 4. Therefore only \(x=-2\) has a finite two-sided limit among the listed inputs.

Answer

At \(x=-2\), the finite two-sided limit exists and equals \(1\). At \(x=1\), the two-sided limit does not exist. At \(x=3\), no finite two-sided limit exists.
54267012
The graph shows \(r\) near \(x=1\). Among \(\delta=0.5\), \(\delta=1\), and \(\delta=1.5\), choose the largest value that the graph supports for the statement: if \(0<|x-1|<\delta\), then \(|r(x)-2|<1\).
Figure for problem 542670

Hints

- Translate the output condition into a horizontal band on the graph. - For each candidate, inspect the full punctured interval around \(x=1\). - Choose the largest listed interval that remains inside the required band.

Solution

1. The condition \(|r(x)-2|<1\) means the graph must stay between y-values \(1\) and \(3\). 2. Within \(1\) unit of \(x=1\), the displayed graph stays inside that horizontal band. 3. At inputs about \(1.5\) units from \(1\), the graph rises above \(3\), so \(\delta=1.5\) is not supported. 4. The largest listed value supported by the graph is \(\delta=1\).

Answer

\(\delta=1\)
55592112
The graph and table represent the same function \(g\) near \(x=2\). The table values are rounded to the nearest whole number. <table><tr><th>\(x\)</th><td>\(1.99\)</td><td>\(1.999\)</td><td>\(2.001\)</td><td>\(2.01\)</td></tr><tr><th>Displayed \(g(x)\)</th><td>\(3\)</td><td>\(3\)</td><td>\(3\)</td><td>\(3\)</td></tr></table> a) What approximate limit does the rounded table suggest? b) Use the graph to give a more precise estimate of \(\lim_{x\to2}g(x)\). c) The table appears to suggest \(3\), while the graph suggests a different decimal value. Explain why the two representations are compatible rather than contradictory, and state the best estimate of the limit.
Figure for problem 555921

Hints

- Treat the table's displayed precision as part of the evidence. - Use the graph's quarter-unit grid to read a more precise limiting height. - Ask what numbers would all display as \(3\) when rounded to the nearest whole number.

Solution

1. Because all displayed table outputs round to \(3\), the table supports only the coarse estimate that the limit is near \(3\). 2. From the graph, the function approaches y-value \(3.25\) as \(x\to2\), so the graph gives the more precise estimate \(3.25\). 3. A value near \(3.25\) rounds to \(3\) to the nearest whole number. Therefore the table's coarse display is compatible with the graph's more precise estimate. 4. The best estimate from the two representations is \(\lim_{x\to2}g(x)\approx3.25\).

Answer

a) The table suggests a limit near \(3\). b) The graph suggests \(3.25\). c) These agree once the table's whole-number rounding is considered; the best estimate is \(\lim_{x\to2}g(x)\approx3.25\).

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