## anonymous 5 years ago EVALUATE sqrt{X ^{2}+2}/ X-1 AS X GOES TO INFINITY

1. anonymous

$\lim_{x \rightarrow \infty} \sqrt{x^{2}+2}/(x-1)$ Factor out an x^2 in the numerator and an x in the denominator. This gets you the following: $\lim_{x \rightarrow \infty}\sqrt{x^{2}(1+(2/x^{2}))}/x(1-(1/x))$ $\lim_{x \rightarrow \infty}x\sqrt{(1+(2/x^{2}))}/x(1-(1/x))$ $\lim_{x \rightarrow \infty}\sqrt{(1+(2/x^{2}))}/(1-(1/x))$ Notice that 2/x^2 and 1/x go to zero as x -> infinity because as the denominator of a fraction gets larger, the value of the fraction gets smaller. So, $\lim_{x \rightarrow \infty}\sqrt{(1+(2/x^{2}))}/(1-(1/x))0 =\sqrt{(1+0)}/(1-0)$ And the square root of 1 over 1 equals 1 so, $\lim_{x \rightarrow \infty} \sqrt{x^{2}+2}/(x-1) = 1$

2. anonymous

THANK YOU