Question

number 8 Investigate the following limits using graph and table use X values: -0.03,-0.02,-0.01,0,0.01,0.02 tell what the limit is if the limit doesn't exist then explain why

Answer 1

Analyze the existence of the following limit:

`\lim _(x\to2)x^2-(2|x-2|)/(x-2)`

To better analyze the limit, we'll use some properties to isolate the required part from the others:

`\lim _(x\to2)x^2-(2|x-2|)/(x-2)=\lim _(x\to2)x^2-\lim _(x\to2)(2|x-2|)/(x-2)`

The first limit is easy to calculate, it's just to replace the value of x. The second limit will be further worked on:

`2^2-2\lim _(x\to2)(|x-2|)/(x-2)=4-2\lim _(x\to2)(|x-2|)/(x-2)`

Now focus on this part:

`\lim _(x\to2)(|x-2|)/(x-2)`

We'll approach the value of x=2 by using the set

X={-0.03, -0.02, -0.01, 0, 0.01, 0.02, 0.03}

Note these values are the infinitesimal approaches to the required value of x=2, thus the values of x to use are x={1.97, 1.98, 1.99, 2, 2.01, 2.02, 2.03}

`(|1.97-2|)/(1.97-2)=(|-0.03|)/(-0.03)=(0.03)/(-0.03)=-1`
`(|1.98-2|)/(1.98-2)=(|-0.02|)/(-0.02)=(0.02)/(-0.02)=-1`
`(|1.99-2|)/(1.99-2)=(|-0.01|)/(-0.01)=(0.01)/(-0.01)=-1`

The value of x=2 cannot be used because it would produce the division 0/0 and it's undefined.

`(|2.01-2|)/(2.01-2)=(|0.01|)/(0.01)=(0.01)/(0.01)=1`
`(|2.02-2|)/(2.02-2)=(|0.02|)/(0.02)=(0.02)/(0.02)=1`
`(|2.03-2|)/(2.03-2)=(|0.03|)/(0.03)=(0.03)/(0.03)=1`

We can see this limit results in -1 for the negative values of X and 1 for the positive values of X. Since the limits are not equal, the limit does not exist