If the function f(x) = frac{sqrt{1+x}-1}{x} i | vedclass

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(C) For a function $f(x)$ to be continuous at $x=0$,the condition $\lim_{x \rightarrow 0} f(x) = f(0)$ must hold.We calculate the limit as follows:$f(

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$\frac{1}{2}$

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(C) For a function $f(x)$ to be continuous at $x=0$,the condition $\lim_{x \rightarrow 0} f(x) = f(0)$ must hold.We calculate the limit as follows:$f(0) = \lim_{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$To evaluate this limit,we rationalize the numerator:$f(0) = \lim_{x \rightarrow 0} \frac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{x(\sqrt{1+x}+1)}$$f(0) = \lim_{x \rightarrow 0} \frac{(1+x)-1}{x(\sqrt{1+x}+1)}$$f(0) = \lim_{x \rightarrow 0} \frac{x}{x(\sqrt{1+x}+1)}$$f(0) = \lim_{x \rightarrow 0} \frac{1}{\sqrt{1+x}+1}$Substituting $x=0$:$f(0) = \frac{1}{\sqrt{1+0}+1} = \frac{1}{1+1} = \frac{1}{2}$

If the function $f(x) = \frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$,then $f(0) = $

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