Question

Find the value of the following limit:
$\underset{x\to 0}{lim}\frac{(1+x{)}^5-(1+5x)}{x^2+x^5}$

Answer 1

$\underset{x\to 0}{lim}\frac{(1+x{)}^5-(1+5x)}{x^2+x^5}$

Applying L Hospital in both question here $\frac{0}{0}$ from occurs

$=\underset{x\to 0}{lim}\frac{\frac{d}{dx}\left[\right(1+x{)}^5-(1+5x)]}{\frac{d}{dx}[x^2+x^5]}$

$=\underset{x\to 0}{lim}\frac{4(1+x{)}^4-5}{2x+5x^4}$

Again $\frac{0}{0}$ from

$=\underset{x\to 0}{lim}\frac{\frac{d}{dx}\left(5\right(1+x{)}^4-5)}{\frac{d}{dx}(2x+5x^4)}$

$=\underset{x\to 0}{lim}\frac{20(1+x{)}^3}{2+30x^3}=\frac{20}{2}=10$.