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  • Evaluate the limit of the following:<img alt="\lim_{x\rightarrow \infty }\left ( \frac{x-25}{x+16} \right )^{x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%20%5Ci

Evaluate the limit of the following:\lim_{x\rightarrow \infty }\left ( \frac{x-25}{x+16} \right )^{x}

Option: 1

\sqrt{e}


Option: 2

e^{-41}


Option: 3

e^{41}


Option: 4

e


Answers (1)

best_answer

Given: \lim_{x\rightarrow \infty }\left ( \frac{x-25}{x+16} \right )^{x}

= \lim_{x\rightarrow \infty }\left ( \frac{x-\frac{25}{x}}{1+\frac{16}{x}} \right )^{x}
= \frac{\lim_{x\rightarrow \infty }\left ( 1-\frac{25}{x} \right )^{x}}{\lim_{x\rightarrow \infty }\left ( 1+\frac{16}{x} \right )^{x}}

Multiply the exponent in the numerator by -25, and denominator by 16,  then the equation becomes,

= \frac{\lim_{x\rightarrow \infty }\left [ \left ( 1-\frac{25}{x} \right )^{\frac{-x}{25}} \right ]^{-25}}{\lim_{x\rightarrow \infty }\left [ \left ( 1+\frac{16}{x} \right )^{\frac{x}{16}} \right ]^{16}}

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

= \frac{e^{-25}}{e^{16}}
= e^{-41}

Therefore, \lim_{x\rightarrow \infty }\left ( \frac{x-25}{x+16} \right )^{x}= e^{-41}.




 

Posted by

Shailly goel

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