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Prove that if $\lim_{x\to a}\,g(x)=0$ then the limit $\lim_{x\to a}\,\frac{1}{g(x)}$ does not exist.

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Lesson Parent:

Function Limits III: The Gory Details

Problem:

Prove that if $\lim_{x\to a}\,g(x)=0$ then the limit $\lim_{x\to a}\,\frac{1}{g(x)}$ does not exist.

Answer:

the limit $\lim_{x\to a}\,\frac{1}{g(x)}$ does not exist.

Solution:

Recall that for a limit to exist we must prove that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>. We don't know what the limit is at this point so lets just assign it a variable $K$ . So for this particular problem we need to show that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>. The main problem here is that as $x$ gets closer and closer to $a$ then <Sign in to see all the formulas> gets larger and larger. In fact, we will show that there are values for $x$ that make <Sign in to see all the formulas> rather than <Sign in to see all the formulas>!

Now let's do some quick algebra. Since we want to show that

<Sign in to see all the formulas>

we know that

<Sign in to see all the formulas>

and therefore

<Sign in to see all the formulas>

The important result that we will use shortly is that

<Sign in to see all the formulas>

We will show that there are values for

$x$ that make <Sign in to see all the formulas> when we always want <Sign in to see all the formulas>.

Because

<Sign in to see all the formulas>

we can choose an

<Sign in to see all the formulas>

and know there is a <Sign in to see all the formulas> such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>.

Now let's choose

<Sign in to see all the formulas>

and assume that we are at a value of

$x$ such that <Sign in to see all the formulas>. As mentioned above, we know that <Sign in to see all the formulas> and therefore

<Sign in to see all the formulas>

Remember that we want to show that for all values of

$x$ such that <Sign in to see all the formulas> then <Sign in to see all the formulas>. However, we found values of

$x$ where <Sign in to see all the formulas>. In particular, when

$x$ is between <Sign in to see all the formulas> and <Sign in to see all the formulas> then <Sign in to see all the formulas>. Since <Sign in to see all the formulas> then there are values of

$x$ in <Sign in to see all the formulas> where <Sign in to see all the formulas> is not true.

Recall that we did not know the limit and decided to just denote it by $K$ . But we have proven that for any real number $K$ there are $x$ values in the range of <Sign in to see all the formulas> where <Sign in to see all the formulas> is not true. So for every real number $K$ there will be some values of $x$ close enough to $a$ to make <Sign in to see all the formulas> not true. Therefore, there is no real number K such that when <Sign in to see all the formulas> then <Sign in to see all the formulas> is always true. The limit $K$ does not exist.