Question

Find the limit :-
$\underset{x\to \infty }{lim}{\left\{\frac{x^2+2x+3}{2x^2+x+5}\right\}}^{\frac{3x-2}{3x+2}}$

Answer 1

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

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

$\underset{x\to \infty }{lim}\frac{1}{x}=0\underset{x\to \infty }{lim}\frac{1}{x^2}=0$

$=\left(\frac{1+0+0}{2+0+2}\right)$

$=\left(\frac{1}{2}\right)$

$=1/2$