Limit at Infinity| Differential and Integral Calculus| Feliciano and Uy| Exercise 1.4, Problem 5|

Evaluate

\lim\limits_{x\to \infty }\left(\frac{8x-5}{\sqrt{4x^2+3}}\right)

SOLUTION:

Divide by the highest denominator power 

\lim\limits_{x\to \infty }\left(\frac{8x-5}{\sqrt{4x^2+3}}\right)=\lim\limits_{x\to \infty }\left(\frac{8x-5}{\sqrt{4x^2+3}}\cdot \frac{\frac{1}{x}}{\frac{1}{x}}\right)

=\lim\limits_{x\to \infty }\left(\frac{\frac{8x}{x}-\frac{5}{x}}{\sqrt{\frac{4x^2}{x^2}+\frac{3}{x^2}}}\right)

=\lim\limits_{x\to \infty }\left(\frac{8-\frac{5}{x}}{\sqrt{4+\frac{3}{x^2}}}\right)

=\frac{8-0}{\sqrt{4+0}}

=\frac{8}{2}

=4

 

 

 

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