Question

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

Answer 1

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

It becomes $\frac{0}{0}$ form.

On multiplying numerator and denominator by $x$, we get

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

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

Where $x\to \infty$, then $\frac{1}{x}\to 0$

$=\frac{3+4\left(0\right)}{5+6\left(0\right)}=\frac{3}{5}$

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