Question

Calculate the following limits.
$\underset{n\to \infty }{lim}\frac{3n^4+2n^3+3}{12n^4-7n^2+n},nϵN.$

Answer 1

$\underset{n\to \infty }{lim}\frac{3n^4+2n^3+3}{12n^4-7n^2+n}$

$=\underset{n\to \infty }{lim}\frac{3+\frac{2n^3}{n^4}+\frac{3}{n^4}}{12-\frac{7n^2}{n^4}+\frac{n}{n^4}}$

$=\underset{n\to \infty }{lim}\frac{3+\frac{2}{n}+\frac{3}{n^4}}{12-\frac{7}{n^2}+\frac{1}{n^3}}$

$=\frac{3+\frac{1}{\infty }+\frac{3}{\infty }}{12-\frac{7}{\infty }+\frac{1}{\infty }}$

$=\frac{1}{4}$