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Prove that function limits are unique.

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Lesson Parent:

Problem:

If
\begin{equation}
\lim_{x\to a}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a}\,f(x)=K
\end{equation}
then $L$ must equal $K$.

Answer:

It is true that function limits are unique.

Solution:

The proof is by contradiction. We assume that <Sign in to see all the formulas> and show that it leads to the contradiction <Sign in to see all the formulas>. Since a real number cannot be less than itself this is a contradiction.

Since

Now assume that we are at an $x$ where <Sign in to see all the formulas> where <Sign in to see all the formulas>. Now we do some simple algebra:

where we have used the triangle inequality for the second line. We have thus proven that <Sign in to see all the formulas> and since <Sign in to see all the formulas> we have our contradiction of <Sign in to see all the formulas>.