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  • Find the limit: <img alt="\lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%200%7D%20%5Cfrac%7B%5Csqrt%7Bx&amp

Find the limit: \lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x}

Option: 1

\frac{1}{2}


Option: 2

\frac{1}{4}


Option: 3

\frac{1}{6}


Option: 4

\frac{1}{8}


Answers (1)

best_answer

\begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x}=\lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x+9-9} \\ = & \lim _{x \rightarrow 0}\left(\frac{\sqrt{x+9}-3}{x+9-9}\right) \cdot\left(\frac{\sqrt{x+9}+3}{\sqrt{x+9}+3}\right) \\ \end{aligned}\begin{aligned} = & \lim _{x \rightarrow 0} \frac{(x+9-9)}{(x+9-9)(\sqrt{x+9}+3)} \\ = & \lim _{x \rightarrow 0} \frac{1}{\sqrt{x+9}+3} \\ = & \frac{1}{\sqrt{9}+3}=\frac{1}{6} \end{aligned}

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Rakesh

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