Question

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

Answer 1

${lim}_{x\to \infty }\frac{3x^{-1}+4x^{-2}}{5x^{-1}+6x^{-2}}\left[Itisin\left[\frac{0}{0}form\right]\right]$

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

$={lim}_{x\to \infty }\frac{\frac{1}{x}(3+\frac{4}{x})}{\frac{1}{x}(5+\frac{6}{x})}={lim}_{x\to \infty }\frac{3+\frac{4}{x}}{5+\frac{6}{x}}$

$asx\to \infty ,\frac{1}{x}\to 0$

$={lim}_{x\to \infty }\frac{(3+0)}{(5+0)}=\frac{3}{5}$