Question

use the graph of the function f shown to estimate the indicated quantities to the nearest integer. complete parts a through e. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. f(2) = 0 b. the value does not exist. e. is f continuous at x = 2? yes no

Explanation:

Part e: Is \( f \) continuous at \( x = 2 \)?

To determine if a function \( f(x) \) is continuous at \( x = a \), three conditions must be met:

1. \( f(a) \) is defined.
2. \( \lim_{x \to a} f(x) \) exists (i.e., the left - hand limit \( \lim_{x \to a^{-}} f(x) \) and the right - hand limit \( \lim_{x \to a^{+}} f(x) \) are equal).
3. \( \lim_{x \to a} f(x)=f(a) \).

Step 1: Check if \( f(2) \) is defined

From part (a) (or the graph), we know that \( f(2) = 0 \), so \( f(2) \) is defined.

Step 2: Check if \( \lim_{x \to 2} f(x) \) exists

• Left - hand limit (\( x \to 2^{-} \)): As \( x \) approaches 2 from the left (values less than 2), we look at the graph. The \( y \) - value that the function approaches as \( x \) gets closer to 2 from the left is 0 (by estimating from the graph).
• Right - hand limit (\( x \to 2^{+} \)): As \( x \) approaches 2 from the right (values greater than 2), we look at the graph. The \( y \) - value that the function approaches as \( x \) gets closer to 2 from the right is also 0 (by estimating from the graph).

Since the left - hand limit and the right - hand limit are equal (\( \lim_{x \to 2^{-}} f(x)=\lim_{x \to 2^{+}} f(x) = 0 \)), \( \lim_{x \to 2} f(x) \) exists and \( \lim_{x \to 2} f(x)=0 \).

Step 3: Check if \( \lim_{x \to 2} f(x)=f(2) \)

We know that \( \lim_{x \to 2} f(x) = 0 \) and \( f(2)=0 \). So, \( \lim_{x \to 2} f(x)=f(2) \).

Since all three conditions for continuity are satisfied, the function \( f \) is continuous at \( x = 2 \).

Answer:

Yes