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