Evaluate the Given limit:

$\lim _{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}$

Put $x+1=y$ so that $y \rightarrow 1$ as $x \rightarrow 0$

Accordingly, $\lim _{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}=\lim _{y \rightarrow 1} \frac{y^{5}-1}{y-1}$

$=\lim _{y \rightarrow 1} \frac{y^{5}-1^{5}}{y-1}$

$=5 \cdot 1^{5-1}$ $\left[\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}\right]$

$=5$

$\therefore \lim _{x \rightarrow 0} \frac{(x+5)^{5}-1}{x}=5$