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Prove that $\begin{equation}\lim_{x\to a}\,x^n=a^n.\end{equation}$

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Lesson Parent:

Problem:

Prove that $\begin{equation}\lim_{x\to a}\,x^n=a^n\end{equation}$ where $n$ is any positive integer.

Answer:

It is true that $\begin{equation}\lim_{x\to a}\,x^n=a^n\end{equation}$ where $n$ is any positive integer.

Solution:

We will prove this by induction.

First, we need to show that for some starting value of $n$ this is true. We have done this both for $n=1$ and $n=2$ .

Next, we need to assume that it is true for some value of $n$ and then prove that it is true for $n+1$ . So let's assume that

Next, we need to prove that

Because

we have put it into the following form and therefore

This completes the proof.