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.

SAMPLE SOLUTION

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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

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we know that

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and therefore

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The important result that we will use shortly is that

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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

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we can choose an

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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

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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.