Calculate a limit of exponential function Calculate this limit:
$$
\lim_{x \to \infty } = \left(\frac{1}{5} + \frac{1}{5x}\right)^{\frac{x}{5}}
$$
I did this:
$$
\left(\frac{1}{5}\right)^{\frac{x}{5}}\left[\left(1+\frac{1}{x}\right)^{x}\right]^\frac{1}{5}
$$
$$
\left(\frac{1}{5}\right)^{\frac{x}{5}}\left(\frac{5}{5}\right)^\frac{1}{5}
$$
$$
\left(\frac{1}{5}\right)^{\frac{x}{5}}\left(\frac{1}{5}\right)^\frac{5}{5}
$$
$$
\lim_{x \to \infty } = \left(\frac{1}{5}\right)^\frac{x+5}{5}
$$
$$
\lim_{x \to \infty } = \left(\frac{1}{5}\right)^\infty = 0
$$
Now I checked on Wolfram Alpha and the limit is $1$
What did I do wrong? is this the right approach? is there an easier way?:)
Edit:
Can someone please show me the correct way for solving this? thanks.
Thanks

$$\lim_{x\to\infty}\left(\frac15\right)^x=0$$
$$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$$
