Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (c)/Solution 1

If you try to evaluate the limit directly, you end up with $\infty /\infty$, which means we cannot conclude anything for the moment, more work needs to be done. Here we have

${\begin{aligned}\lim _{x\to \infty }{\frac {2x}{\sqrt {9x^{2}+x-1}}}&=\lim _{x\to \infty }{\frac {2}{\frac {\sqrt {9x^{2}+x-1}}{x}}}\\&=\lim _{x\to \infty }{\frac {2}{\frac {\sqrt {9x^{2}+x-1}}{\sqrt {x^{2}}}}}\\&=\lim _{x\to \infty }{\frac {2}{\sqrt {\frac {9x^{2}+x-1}{x^{2}}}}}\\&=\lim _{x\to \infty }{\frac {2}{\sqrt {9+{\frac {1}{x}}-{\frac {1}{x^{2}}}}}}\\&={\frac {2}{\sqrt {9}}}\\&={\frac {2}{3}}.\end{aligned}}$