Question

please solve the limit .. ans is 1/4√3 i need step by step solving . i will mark them as brainliest​

Answer 1

Answer:

1/(4√3)

Step-by-step explanation:

Directly substituting the limits results (0/0) which is an indeterminate quantity. Therefore we have to choose different method.

We will solve this question using the method of rationalisation.

$\implies \lim \limits_{x \to 0} \dfrac{ \sqrt{3 + x} - \sqrt{3 - x} }{ {x}^{2} + 4x}$

${ \implies \lim \limits_{x \to 0} \dfrac{ \sqrt{3 + x} - \sqrt{3 - x} }{x(x + 4)} \times \dfrac{ \sqrt{3 + x} + \sqrt{3 - x} }{ \sqrt{3 + x} + \sqrt{3 - x} } }$

${ \implies \lim \limits_{x \to 0} \dfrac{ (\sqrt{3 + x})^{2} - (\sqrt{3 - x} )^{2} }{x(x + 4)(\sqrt{3 + x} + \sqrt{3 - x})} }$

${ \implies \lim \limits_{x \to 0} \dfrac{3 + x - 3 + x}{x(x + 4)(\sqrt{3 + x} + \sqrt{3 - x})} }$

${ \implies \lim \limits_{x \to 0} \dfrac{2x}{x(x + 4)(\sqrt{3 + x} + \sqrt{3 - x})} }$

${ \implies \lim \limits_{x \to 0} \dfrac{2}{(x + 4)(\sqrt{3 + x} + \sqrt{3 - x})} }$

${ \implies \dfrac{2}{(0+ 4)(\sqrt{3 + 0} + \sqrt{3 - 0})} }$

${ \implies \dfrac{2}{4(2\sqrt{3})} }$

$\implies \dfrac{1}{4\sqrt{3}}$