QuestionFind the largest $\delta$ for $\lim _{x \rightarrow 2}\left(x^{3}-5 x+5\right)=3$ with $\varepsilon=0.2$ and $\varepsilon=0.1$ . Round to four decimal places.

Studdy Solution

STEP 1

Assumptions1. We are given the limit $\lim{x \rightarrow}\left(x^{3}-5 x+5\right)=3$ . . We are asked to find the largest possible values of $\delta$ for $\varepsilon=0.$ and $\varepsilon=0.1$ .
3. The definition of limit we are using is the ε-δ definition, which states that for every positive number ε, there exists a positive number δ such that if $0<|x-a|<\delta$ , then $|f(x)-|<\varepsilon$ .
4. In this case, $a=$ , $f(x)=x^{3}-5 x+5$ , and $=3$ .

STEP 2

First, we need to find the expression for $|f(x)-|$ . $|f(x)-|=|x^{}-5 x+5-|$

STEP 3

implify the expression.
$|f(x)-|=|x^{3}-5 x+2|$

STEP 4

We are given that $|f(x)-|<\varepsilon$ . So, we have two cases to consider $x^{3}- x+2<\varepsilon$ and $x^{3}- x+2>-\varepsilon$ .

STEP 5

Let's solve the first inequality $x^{3}-5 x+2<\varepsilon$ for $x$ when $\varepsilon=0.2$ .
$x^{3}-5 x+2<0.2$

STEP 6

This is a cubic inequality, and it's not easy to solve algebraically. So, we'll use a graphing calculator to find the roots of the equation $x^{3}-5 x+2=0.2$ .

STEP 7

The roots of the equation are approximately $x=1.8077$ and $x=2.1912$ .

STEP 8

Since we are interested in the limit as $x$ approaches $2$ , we take the smaller root $x=1.8077$ and calculate $|x-2|$ to find $\delta$ .
$\delta=|x-2|=|1.8077-2|=0.1923$

STEP 9

Now, let's solve the second inequality $x^{3}-5 x+2>-\varepsilon$ for $x$ when $\varepsilon=.2$ .
$x^{3}-5 x+2>-.2$

STEP 10

Again, we'll use a graphing calculator to find the roots of the equation $x^{3}-5 x+2=-0.2$ .

STEP 11

The roots of the equation are approximately $x=.8088$ and $x=.190$ .

STEP 12

Again, since we are interested in the limit as $x$ approaches $2$ , we take the smaller root $x=.8088$ and calculate $|x-2|$ to find $\delta$ .
$\delta=|x-2|=|.8088-2|=0.1912$

STEP 13

The largest possible value of $\delta$ for $\varepsilon=0.2$ is the maximum of the two $\delta$ values we found, which is $\delta=0.1923$ .

STEP 14

Now, let's repeat the process for $\varepsilon=0.$ . First, solve the inequality $x^{3}- x+2<0.$ for $x$ .

STEP 15

Use a graphing calculator to find the roots of the equation $x^{3}-5 x+2=0.$ .

STEP 16

The roots of the equation are approximately $x=.904$ and $x=2.0958$ .

STEP 17

Calculate $\delta=|x-2|$ for the smaller root $x=.904$ .
$\delta=|x-2|=|.904-2|=0.0959$

STEP 18

Now, solve the inequality $x^{3}-5 x+2>-0.$ for $x$ .

STEP 19

Use a graphing calculator to find the roots of the equation $x^{3}-5 x+=-.1$ .

STEP 20

The roots of the equation are approximately $x=.904$ and $x=.0957$ .

STEP 21

Calculate $\delta=|x-|$ for the smaller root $x=1.904$ .
$\delta=|x-|=|1.904-|=0.0958$

STEP 22

The largest possible value of $\delta$ for $\varepsilon=0.1$ is the maximum of the two $\delta$ values we found, which is $\delta=0.0959$ .
So, the largest possible values of $\delta$ for $\varepsilon=0.$ and $\varepsilon=0.1$ are $\delta=0.192$ and $\delta=0.0959$ , respectively.
$\begin{array}{ll} \varepsilon=0. & \delta=0.192 \\ \varepsilon=0.1 & \delta=0.0959\end{array}$

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