# What is the best way to solve problem like these ↓ ? (Limits to infinity)

## Problems such as When $x$ approaches positive infinity, find the limit of (3x-1)/(sqrt(x^2-6 $\frac{\sqrt{4 {x}^{2} + 4 x}}{4 x + 1}$ I know that you can solve by substituting values of $x$ as it gets closer to infinity, but is there a way to solve, like factorising, formula, or a general rule?

Jun 22, 2017

As a general rule convert a polynomial $f \left(x\right)$ to $g \left(\frac{1}{x}\right)$
(A) $3$ and (B) $\frac{1}{2}$

#### Explanation:

$L {t}_{x \to \infty} \frac{3 x - 1}{\sqrt{{x}^{2} - 6}}$

• let us divide numerator and denominator by $x$ and we get

$L {t}_{x \to \infty} \frac{3 - \frac{1}{x}}{\sqrt{1 - \frac{6}{x} ^ 2}}$

= $\frac{3}{\sqrt{1}} = 3$
graph{(3x-1)/sqrt(x^2-6) [2.07, 19.55, -0.37, 8.37]}

$L {t}_{x \to \infty} \frac{\sqrt{4 {x}^{2} + 4 x}}{4 x + 1}$

• dividing numerator and denominator by $2 x$ and we get

$L {t}_{x \to \infty} \frac{\sqrt{1 + \frac{2}{x}}}{2 + \frac{1}{2} x}$

= $\frac{1}{2}$
graph{sqrt(4x^2+4x)/(4x+1) [-0.152, 4.218, -0.51, 1.675]}

As a general rule convert a polynomial $f \left(x\right)$ to $g \left(\frac{1}{x}\right)$