# How do you find the limit of  (x^2+2x-1)/(3+3x^2) as x approaches infinity?

May 17, 2018

$\frac{1}{3}$

#### Explanation:

$\text{divide terms on numerator/denominator by } {x}^{2}$

$= \frac{{x}^{2} / {x}^{2} + \frac{2 x}{x} ^ 2 - \frac{1}{x} ^ 2}{\frac{3}{x} ^ 2 + \frac{3 {x}^{2}}{x} ^ 2} = \frac{1 + \frac{2}{x} - \frac{1}{x} ^ 2}{\frac{3}{x} ^ 2 + 3}$

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

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

$= \frac{1 + 0 - 0}{0 + 3} = \frac{1}{3}$