Exercise 4.6.1.
Prove the limit (0.0.2).
Hint.
Use the binomial theorem to express \((x+h)^n\) as a sum of powers of \(x\text{.}\) Some background on the binomial theorem is given below.
Background 4.6.1. The Binomial Theorem.
The factorial of a positive integer \(m\) is \(n!=n(n-1)\cdots 2\cdot 1\text{,}\) and \(0!=1\text{.}\) If \(n\) is a positive integer and \(k\) is a non-negative integer, the binomial coefficient “\(n\) choose \(k\)” is
\begin{equation*}
\binom{n}{k} =\frac{n!}{k!(n-k)!},
\end{equation*}
and it is always a positive integer, giving the number of ways to choose \(k\) objects out of \(n\text{.}\) Using this combinatorial interpretation, we obtain the binomial theorem:
Theorem 4.6.2. The Binomial Theorem.
\begin{equation*}
(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^ky^{n-k}.
\end{equation*}
