Use the graph of $f\left(x\right)$to estimate the value of the limit or state that it does not exist (DNE).
$\lim_{x\rarr0}f\left(x\right)$

• A. $-1$
• B. $0$
• C. $1$
• D. DNE

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to evaluate the limit \( \lim_{x \to 0} f(x) \) using the graph of \( f(x) \). A limit exists if the function approaches a single finite value as \( x \) approaches the given point from both sides.
2. Step 2: Analyze the graph near \( x = 0 \). Observe the behavior of \( f(x) \) as \( x \) approaches 0 from the left (\( x \to 0^- \)) and from the right (\( x \to 0^+ \)).
3. Step 3: Notice that the graph oscillates wildly between \( y = -1 \) and \( y = 1 \) as \( x \) approaches 0. This indicates that \( f(x) \) does not settle to a single value.
4. Step 4: Recall the definition of a limit. For a limit to exist, \( f(x) \) must approach the same value from both sides of \( x = 0 \). Since \( f(x) \) oscillates infinitely, it does not approach a single value.
5. Step 5: Conclude that the limit \( \lim_{x \to 0} f(x) \) does not exist (DNE) because the function does not converge to a single value as \( x \) approaches 0.