Question

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

Answer 1

$$\begin{gathered} \underset{x\to \infty }{\lim }\left[\frac{3x^{-1}+4x^{-2}}{5x^{-1}+6x^{-2}}\right] \\ =\underset{x\to \infty }{\lim }\left[\frac{{\displaystyle \frac{3}{x}}+{\displaystyle \frac{4}{x^2}}}{{\displaystyle \frac{5}{x}}+{\displaystyle \frac{6}{x^2}}}\right] \\ =\underset{x\to \infty }{\lim }\left[\frac{3x+4}{5x+6}\right] \\ \mathrm{Dividing}\text{the numerator and the denominator by x:} \\ \underset{x\to \infty }{\lim }\left[\frac{3+{\displaystyle \frac{4}{x}}}{5+{\displaystyle \frac{6}{x}}}\right]=\frac{3}{5} \end{gathered}$$