Question

Evaluate the limit to infinity
$\underset{x\to \infty }{lim}\frac{{3x}^{-1}+{4x}^{-2}}{{5x}^{-1}+6x^{-2}}=\frac{3}{5}$

Answer 1

$\underset{x\to \infty }{lim}\frac{3x^{-1}+4x^{-2}}{5x^{-1}+6x^{-2}}$

$=\underset{x\to \infty }{lim}\frac{\frac{3}{x}+\frac{4}{x^2}}{\frac{5}{x}+\frac{6}{x^2}}$

$=\underset{x\to \infty }{lim}\frac{\frac{1}{x^2}\left(3x+4\right)}{\frac{1}{x^2}\left(5x+6\right)}$

$=\underset{x\to \infty }{lim}\frac{x\left(3+\frac{4}{x}\right)}{x\left(5+\frac{6}{x}\right)}=\frac{3}{5}$