Evaluate the given limit: $\underset{x\to 0}{\lim }\frac{{\left(x+1\right)}^5-1}{x}$

$\underset{x\to 0}{\lim }\frac{{\left(x+1\right)}^5-1}{x}$
Put $x+1=y$
So, $y\to 1$ as $x\to 0$
$=\underset{y\to 1}{\lim }\frac{y^5-1^5}{y-1}$
$=5\cdot 1^{5-1}$           $\left[\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}=n{a}^{n-1}\right]$
$=5$
$\therefore \underset{x\to 0}{\lim }\frac{{\left(x+5\right)}^5-1}{x}=5$