Question

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

Answer 1

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