Evaluate $\lim_{x\to\infty} (1+\frac{2}{x})^{5x}$ without L'Hopital

Question

I'm trying to evaluate the following limit

$$\lim_{x\to\infty} \left(1+\frac{2}{x}\right)^{5x}$$

I recognize a part of this limit because it resembles the limit for $e$ but I don't know anything other than that. I have no idea where to start. Some hints would be much appreciated.

Thanks!

Answer 1

Write $y = \frac{x}{2}$. Then we want $\displaystyle \lim_{y\rightarrow\infty}\left(1+\frac{1}{y}\right)^{10y}$. Since the function $g(z) = z^{10}$ is continuous, this is $$\lim_{y\rightarrow\infty}g\left(\left(1+\frac{1}{y}\right)^{y}\right) = g\left(\lim_{y\rightarrow\infty}\left(1+\frac{1}{y}\right)^{y}\right) = \left(\lim_{y\rightarrow\infty}\left(1+\frac{1}{y}\right)^y\right)^{10} = e^{10}.$$

Answer 2

Hint: $y:=x/2$, then $5x=10y$ and we have $$\left(1+\frac2x\right)^{5x}=\left(1+\frac1y\right)^{y\cdot 10}\,.$$

Answer 3

Hint: $$ \lim_{x\rightarrow \infty}\left(1+\frac{2}{x}\right)^\frac{x}{2}=e $$