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  • The value of   so that the fanction <img alt="\mathrm{f(x)= \frac{\sqrt{1+x}-\sqrt[3]{1

The value of \mathrm{f(0)}  so that the fanction \mathrm{f(x)= \frac{\sqrt{1+x}-\sqrt[3]{1+x}}{x}} becomes continuous is equal to
 

Option: 1

\frac{1 }{ 6}


Option: 2

\frac{1 }{ 4}


Option: 3

\frac{1 }{ 3}


Option: 4

3


Answers (1)

best_answer

\begin{aligned} &\mathrm{ f(x)=\frac{1}{x}\left[(1+x)^{1 / 2}-(1+x)^{1 / 3}\right] }\\ &\mathrm{= \frac{1}{x}\left[\left(1+\left(\frac{1}{2}\right) x+0\left(x^2\right)\right)-\left(1+\left(\frac{1}{3}\right) x+0\left(x^2\right)\right)^2\right] }\\ &\mathrm{= \frac{1}{x}\left[\left(\frac{1}{6}\right) x+0\left(x^2\right)\right]=\frac{1}{6}+0(x) . }\\ &\mathrm{ \lim _{x \rightarrow 0} f(x)=\frac{1}{6} .}\\ \end{aligned}

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