1. Limits and Continuity

Introduction to Limits

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

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

$-1$

$0$

$1$

DNE

Verified step by step guidance

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.

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^+ $).

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.

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.

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.