Question

Find the value of the following limit:
$\underset{x\to \infty }{lim}(\frac{3x}{5x-1}-\frac{2x^2+1}{x^2+2x-1})$

Answer 1

$\underset{x\to \infty }{lim}(\frac{3x}{5x-1}\times \frac{2x^2+1}{x^2+2x-1})$

$=\underset{x\to \infty }{lim}(\frac{x\left(3\right)}{x(5-\frac{1}{x})}\times \frac{x^2(2+\frac{1}{x^2})}{x^2(1+\frac{2}{x}-\frac{1}{x^2})})$

$=\frac{3}{5-\frac{1}{\infty }}\times \frac{2+\frac{1}{\infty }}{1+\frac{2}{\infty }-\frac{1}{\infty }}$

$=\frac{3}{5}\times \frac{2}{1}[\frac{1}{\infty }=0]$

$=\frac{6}{5}$.