Questions: The function f in the figure satisfies lim x -> 2 f(x)=2. Determine the largest value of δ>0 satisfying each statement. a. If 0<x-2<δ, then f(x)-2<1/2. b. If 0<x-2<δ, then f(x)-2<1/4. a. δ=1 (Simplify your answer.) b. δ= (Simplify your answer.)

The function f in the figure satisfies lim x -> 2 f(x)=2. Determine the largest value of δ>0 satisfying each statement.
a. If 0<x-2<δ, then f(x)-2<1/2.
b. If 0<x-2<δ, then f(x)-2<1/4.
a. δ=1 (Simplify your answer.)
b. δ= (Simplify your answer.)

Transcript text: The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=2$. Determine the largest value of $\delta>0$ satisfying each statement. a. If $0<|x-2|<\delta$, then $|f(x)-2|<\frac{1}{2}$. b. If $0<|x-2|<\delta$, then $|f(x)-2|<\frac{1}{4}$. a. $\delta=1$ (Simplify your answer.) b. $\delta=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

To determine the largest value of $\delta > 0$ for each statement, we need to analyze the behavior of the function $f(x)$ as $x$ approaches 2. Specifically, we need to find the range of $x$ values around 2 such that the function $f(x)$ stays within a specified distance from 2.

Part (a)

For the first statement, we need to find the largest $\delta$ such that if $0 < |x - 2| < \delta$, then $|f(x) - 2| < \frac{1}{2}$.

Part (b)

For the second statement, we need to find the largest $\delta$ such that if $0 < |x - 2| < \delta$, then $|f(x) - 2| < \frac{1}{4}$.

Step 1: Understanding the Problem

We are given a function $ f $ and the limit $ \lim_{x \to 2} f(x) = 2 $. We need to determine the largest value of $ \delta > 0 $ such that the function $ f $ satisfies certain conditions around $ x = 2 $.

Step 2: Analyzing Condition (a)

For condition (a), we need to find the largest $ \delta $ such that if $ 0 < |x - 2| < \delta $, then $ |f(x) - 2| < \frac{1}{2} $.

Given: \[ \lim_{x \to 2} f(x) = 2 \]

This means for any $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that if $ 0 < |x - 2| < \delta $, then $ |f(x) - 2| < \epsilon $.

For $ \epsilon = \frac{1}{2} $: \[ |f(x) - 2| < \frac{1}{2} \]

From the problem statement, it is given that $ \delta = 1 $ satisfies this condition.

Step 3: Analyzing Condition (b)

For condition (b), we need to find the largest $ \delta $ such that if $ 0 < |x - 2| < \delta $, then $ |f(x) - 2| < \frac{1}{4} $.

Given: \[ \lim_{x \to 2} f(x) = 2 \]

For $ \epsilon = \frac{1}{4} $: \[ |f(x) - 2| < \frac{1}{4} \]

We need to determine the largest $ \delta $ that satisfies this condition. Since the limit exists and is equal to 2, we can find such a $ \delta $.

Step 4: Determining the Largest $ \delta $ for Condition (b)

To find the largest $ \delta $ for $ \epsilon = \frac{1}{4} $, we need to analyze the behavior of $ f(x) $ around $ x = 2 $. Without the explicit form of $ f(x) $, we rely on the given limit and the behavior of $ f $ near $ x = 2 $.

Assuming the function $ f $ behaves similarly to condition (a), we can infer that the largest $ \delta $ for $ \epsilon = \frac{1}{4} $ would be smaller than or equal to the $ \delta $ for $ \epsilon = \frac{1}{2} $.