QuestionFind $\delta$ for $\varepsilon=0.1$ in the limit: $\lim _{x \rightarrow 6}-3 x-4=-22$ .

Studdy Solution

STEP 1

Assumptions1. The limit of the function $-3x -4$ as $x$ approaches $6$ is $-22$ . . We need to find a $\delta$ such that for all $x$ satisfying $0 < |x -6| < \delta$ , the absolute difference between $-3x -4$ and $-22$ is less than $\varepsilon=0.1$ .

STEP 2

First, we need to express the absolute difference between $-x -4$ and $-22$ in terms of $x$ .
$|-x -4 - (-22)| = |-x +18| =|x -6|$

STEP 3

We want this absolute difference to be less than $\varepsilon=0.1$ . So, we set up the following inequality $3|x -6| < \varepsilon$

STEP 4

Substitute the given value of $\varepsilon$ into the inequality $3|x -6| <0.1$

STEP 5

To isolate $|x -|$ , divide both sides of the inequality by $3$ $|x -| < \frac{0.1}{3}$

STEP 6

Calculate the value of $\frac{0.1}{3}$ $|x -6| <0.03333...$

STEP 7

The value of $\delta$ that works for $\varepsilon=0.1$ is $\delta =0.03333...$ .
So, for all $x$ satisfying $0 < |x -6| < \delta$ , the absolute difference between $-3x -4$ and $-22$ is less than $\varepsilon=0.1$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms