Chapter 15.1, Problem 5GCE

To determine

To explain:

The limit by using the graphing calculator

Explanation of Solution

Given:

  $f\left(1\right)$ and $\underset{x\to 1}{\lim }f\left(x\right)$

Concept used:

Let $f\left(x\right)$ be a function defined on an interval that contains $x=\infty$ , except possibly at $x=\infty$

Then

  $\underset{x\to \infty }{\lim }f\left(x\right)=L$

Calculation:

Let $f\left(x\right)$ be a function defined on an interval that contains $x=\infty$ , except possibly at $x=\infty$

Then

  $\underset{x\to \infty }{\lim }f\left(x\right)=L$

Counter example:

The corresponding $y-value$

  $f\left(1\right)=2$

If approach an $x-\text{value}$ of $1$ from the left

  $\underset{x\to 1^{-}}{\lim }f\left(x\right)=0$

If approach an $x-\text{value}$ of $1$ from the right

  $\underset{x\to 1^{+}}{\lim }f\left(x\right)=2$

The limit from the left does not equal the limit from the right

  $\underset{x\to 1}{\lim }f\left(x\right)\text{does}\text{not}\text{exist}$

  $f\left(1\right)=2$ but $\underset{x\to 1}{\lim }f\left(x\right)\text{does}\text{not}\text{exist}$