Question

pls evaluate this limit​

Answer 1

GIVEN :–

$\\ \implies \bf \lim_{x \to0} \: \dfrac{ {(x + 1)}^{5} - 1}{x}$

TO FIND :–

• Value of limit = ?

SOLUTION :–

• Let –

$\\ \implies \bf P = \lim_{x \to0} \: \dfrac{ {(x + 1)}^{5} - 1}{x}$

• Now put the limit –

$\\ \implies \bf P = \dfrac{ {(0+ 1)}^{5} - 1}{0}$

$\\ \implies \bf P = \dfrac{ {(1)}^{5} - 1}{0}$

$\\ \implies \bf P = \dfrac{1 - 1}{0}$

$\\ \implies \bf P = \dfrac{0}{0}$

• It's an undefined form. Let use L'HOSPITAL rule –

$\\ \implies \bf P = \lim_{x \to0} \: \dfrac{ {5(x + 1)}^{4} - 1}{1}$

• Now put the limit –

$\\ \implies \bf P = \dfrac{ {5(0 + 1)}^{4} - 1}{1}$

$\\ \implies \bf P = \dfrac{ {5(1)}^{4} - 1}{1}$

$\\ \implies \bf P = \dfrac{5- 1}{1}$

$\\ \implies \bf P = \dfrac{4}{1}$

$\\ \implies \large{ \boxed{ \bf P =4}}$

• Hence , The value of limit –

$\\ \implies \large { \boxed{\bf \lim_{x \to0} \: \dfrac{ {(x + 1)}^{5} - 1}{x} = 4}}$