find the limit. use lhospitals rule if appropriate. if there is a more elementary method, lim (1 + a/x)^(bx)…

Explanation:

Step1: Recall the standard limit form

We know that $\lim_{t\rightarrow\infty}(1 + \frac{1}{t})^t=e$. Let $t=\frac{x}{a}$, then $x = at$. As $x\rightarrow\infty$, $t\rightarrow\infty$. The given limit $\lim_{x\rightarrow\infty}(1+\frac{a}{x})^{bx}$ can be rewritten.

Step2: Rewrite the limit

Substitute $t=\frac{x}{a}$ into the limit: [ \begin{align*} \lim_{x\rightarrow\infty}(1 + \frac{a}{x})^{bx}&=\lim_{t\rightarrow\infty}(1+\frac{1}{t})^{b(at)}\ &=\lim_{t\rightarrow\infty}[(1 + \frac{1}{t})^{t}]^{ab} \end{align*} ]

Step3: Apply the standard - limit result

Since $\lim_{t\rightarrow\infty}(1 + \frac{1}{t})^t = e$, we have $\lim_{t\rightarrow\infty}[(1 + \frac{1}{t})^{t}]^{ab}=e^{ab}$.

Answer:

$e^{ab}$