How do you find the limit (x+x^-2)/(2x+x^-2) as x->oo?

Mar 8, 2018

The limit is $= \frac{1}{2}$

Explanation:

Let's rewrite the function

$y = \frac{x + {x}^{-} 2}{2 x + {x}^{-} 2} = \frac{x + \frac{1}{x} ^ 2}{2 x + \left(\frac{1}{x} ^ 2\right)}$

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

$= \frac{{x}^{3} \left(1 + \frac{1}{x} ^ 3\right)}{{x}^{3} \left(2 + \frac{1}{x} ^ 3\right)}$

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

$L i {m}_{x \to \infty} \left(\frac{1}{x} ^ 3\right) = 0$

Therefore,

$L i {m}_{x \to \infty} \left(y\right) = L i {m}_{x \to \infty} \left(\frac{1 + \frac{1}{x} ^ 3}{2 + \frac{1}{x} ^ 3}\right) = \frac{1 + 0}{2 + 0} = \frac{1}{2}$