Question

$\underset{n\to \infty }{lim}\frac{\left(n^2+5n+6\right)}{\left(n+4\right)\left(n+5\right)}isequalto$

Answer 1

We have,

${lim}_{x\to \infty }\frac{n^2+5n+6}{(n+4)(n+5)}$

${lim}_{x\to \infty }\frac{n^2+5n+6}{n^2+4n+5n+20}$

${lim}_{x\to \infty }\frac{n^2+5n+6}{n^2+9n+20}$

${lim}_{x\to \infty }\frac{n^2(1+\frac{5}{n}+\frac{6}{n^2})}{n^2(1+\frac{9}{n}+\frac{20}{n^2})}$

${lim}_{x\to \infty }\frac{(1+\frac{5}{n}+\frac{6}{n^2})}{(1+\frac{9}{n}+\frac{20}{n^2})}$

$\Rightarrow \frac{(1+\frac{5}{\infty }+\frac{6}{{\infty }^2})}{(1+\frac{9}{\infty }+\frac{20}{{\infty }^2})}$

$\Rightarrow \frac{(1+0+0)}{(1+0+0)}$

$\Rightarrow \frac{1}{1}$

$\Rightarrow 1$

Hence, this is the answer.