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  • Let be such that the function is <img alt="f\left ( x \right )= 2x" src="https://entrance

Let f:R\rightarrow R be such that the function is f\left ( x \right )= 2x,  evaluate the limit of the following: \lim_{x\rightarrow \infty }\left ( 1+f\left ( x \right ) \right )^{1/x}

Option: 1

\sqrt{e}


Option: 2

0


Option: 3

e^{2}


Option: 4

e


Answers (1)

best_answer

Given: \lim_{x\rightarrow \infty }\left ( 1+f\left ( x \right ) \right )^{1/x}

Substitute the value of function as f\left ( x \right )= 2x,

= \lim_{x\rightarrow \infty}\left ( 1+2x \right )^{1/x}

Multiply and divide the exponent by 2, then the equation becomes,

= {\lim_{x\rightarrow \infty}}\left ( 1+2x \right )^{1/2x\left ( 2 \right )}

Using the property of the limit = {\lim_{x\rightarrow a}}\left ( 1+x \right )^{1/x}= e.
= e^{2}

Therefore, \lim_{x\rightarrow \infty }\left ( 1+f\left ( x \right ) \right )^{1/x}= e^{2}.
 

Posted by

Irshad Anwar

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