Evaluate: mathop {lim }limits_{x to 0} frac{s | vedclass

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Question

(A) To evaluate the limit $\mathop {\lim }\limits_{x 	o 0} \frac{\sqrt{1+x}-1}{x}$,we can rationalize the numerator: $\mathop {\lim }\limits_{x 	o 0

Vedclass · Accepted Answer

$1/2$

Vedclass · Answer

(A) To evaluate the limit $\mathop {\lim }\limits_{x 	o 0} \frac{\sqrt{1+x}-1}{x}$,we can rationalize the numerator: $\mathop {\lim }\limits_{x 	o 0} \frac{\sqrt{1+x}-1}{x} 	imes \frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}$ $= \mathop {\lim }\limits_{x 	o 0} \frac{(1+x)-1}{x(\sqrt{1+x}+1)}$ $= \mathop {\lim }\limits_{x 	o 0} \frac{x}{x(\sqrt{1+x}+1)}$ $= \mathop {\lim }\limits_{x 	o 0} \frac{1}{\sqrt{1+x}+1}$ Substituting $x = 0$,we get: $\frac{1}{\sqrt{1+0}+1} = \frac{1}{1+1} = \frac{1}{2}$

Evaluate: $\mathop {\lim }\limits_{x \to 0} \frac{\sqrt{1+x}-1}{x}$

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