Menu

Show that $\lim_{h\to 0}\,\frac{|h|}{h}$ does not exist.

Primary tabs

Lesson Parent:

Function Limits VIII: Limits Involving Zero

Problem:

Show that $\lim_{h\to 0}\,\frac{|h|}{h}$ does not exist.

Answer:

It is true that $\lim_{h\to 0}\,\frac{|h|}{h}$ does not exist.

Solution:

<Sign in to see all the formulas>

if and only if

<Sign in to see all the formulas>

We will prove that the limit does not exist by showing that

<Sign in to see all the formulas>

First, we assert that

<Sign in to see all the formulas>

To prove this left-sided limit we have to show that for <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. Since

$h$ is always less than 1, <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. We can choose any <Sign in to see all the formulas> for any <Sign in to see all the formulas> and always have <Sign in to see all the formulas> when <Sign in to see all the formulas>. Therefore we have proven that

<Sign in to see all the formulas>

Next, we assert that

<Sign in to see all the formulas>

To prove this right-sided limit we have to show that for <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. Since

$h$ is always greater than 1, <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. We can choose any <Sign in to see all the formulas> for any <Sign in to see all the formulas> and always have <Sign in to see all the formulas> when <Sign in to see all the formulas>. Therefore we have proven that

<Sign in to see all the formulas>

Since,

<Sign in to see all the formulas>

the function limit

<Sign in to see all the formulas>

does not exist.