Graph of f appears in Figure 10.11. Determine the following limits: (a) lim x→-1 f(x) (b) lim x→0 f(x) (c) lim x→1 f(x)
Understand the Problem
The question asks us to find the limit of the function f(x) as x approaches -1, 0, and 1, given the graph of f(x). We need to visually determine the value that the function approaches from both the left and right sides for each x value.
Answer
(a) $\lim_{x \to -1} f(x) = 1$ (b) $\lim_{x \to 0} f(x) = 0$ (c) $\lim_{x \to 1} f(x)$ does not exist
Answer for screen readers
(a) $\lim_{x \to -1} f(x) = 1$ (b) $\lim_{x \to 0} f(x) = 0$ (c) $\lim_{x \to 1} f(x)$ does not exist
Steps to Solve
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Find $\lim_{x \to -1} f(x)$ Examine the graph as $x$ approaches $-1$ from the left and the right along the x-axis. From the left, the function approaches $1$. From the right, the function approaches $1$. Since the left and right limits are equal, the limit exists and is equal to $1$.
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Find $\lim_{x \to 0} f(x)$ Examine the graph as $x$ approaches $0$ from the left and the right along the x-axis. From the left, the function approaches $0$. From the right, the function approaches $0$. Since the left and right limits are equal, the limit exists and is equal to $0$.
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Find $\lim_{x \to 1} f(x)$ Examine the graph as $x$ approaches $1$ from the left and the right along the x-axis. From the left, the function approaches $1$. From the right, the function approaches $-\infty$. Since the left and right limits are not equal, the limit does not exist.
(a) $\lim_{x \to -1} f(x) = 1$ (b) $\lim_{x \to 0} f(x) = 0$ (c) $\lim_{x \to 1} f(x)$ does not exist
More Information
The limit of a function at a point exists only if the left-hand limit and the right-hand limit at that point are equal. If they are not equal, the limit does not exist.
Tips
A common mistake is to confuse the value of the function at a point with the limit of the function at that point. The limit describes the value the function approaches, not necessarily the actual value of the function at that point. Also, it is important to check both the left and right limits.
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