Question

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

Answer 1

We have,

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

${lim}_{x\to \infty }\frac{\left(x^2\right)(4+\frac{5}{x^2})}{x^2(1+\frac{2}{x}+\frac{3}{x^2})}$

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

$\Rightarrow \frac{4+\frac{5}{\infty }}{(1+\frac{2}{\infty }+\frac{3}{\infty })}$

$\Rightarrow \frac{4+0}{(1+0+0)}$

$\Rightarrow \frac{4}{1}=4$

Hence, this is the answer.