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.
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 \(1\).
