Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 56

Determine the $\displaystyle \lim_{x \to \infty} \left( 1+ \frac{a}{x} \right)^{bx}$. Use L'Hospital's Rule where appropriate. Use some Elementary method if posible. If L'Hospitals Rule doesn't apply. Explain why.

We can rewrite the limit as...
$\displaystyle \lim_{x \to \infty} \left( 1 + \frac{a}{x} \right) = \lim_{x \to \infty} \left( 1 + \frac{1}{\frac{x}{a}}\right)^{ba\left(\frac{x}{a} \right)}$
If we let $\displaystyle u = \frac{x}{a}$, then...


$\displaystyle \lim_{x \to \infty} \left( 1 + \frac{1}{u}\right)^{ba u} = \lim_{x \to \infty} \left[ \left( 1 + \frac{1}{u} \right)^u \right]^{ab}$

Recall that $\displaystyle \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e$

$
\begin{equation}
\begin{aligned}
\lim_{x \to \infty} \left[ \left( 1 + \frac{1}{u} \right)^u\right]^{ab} &= [e]^{ab}\\
\\
&= e^{ab}
\end{aligned}
\end{equation}
$