<s>[INST] Evaulate $\lim_{x\to\infty}\frac{3x^2-36x+12}{5x^2+113x-2}$ Question:  Find the limit, $$\lim_{x\to\infty}\frac{3x^2-36x+12}{5x^2+113x-2}$$
The limit should be $\frac{3}{5}$ since when $x$ approaches infinity since $\frac{3*\infty^2}{5*\infty^2}$ and infinity squared cancels out.  Am I correct?
 [/INST] $\displaystyle\frac{3x^2-36x+12}{5x^2+113x-2}=\displaystyle\frac{3-36\frac{1}{x}+12\frac{1}{x^2}}{5+113\frac{1}{x}-2\frac{1}{x^2}}$(Dividing by numerator and denominator by $x^2$ )
As $\lim_{x\to \infty }(3-36\frac{1}{x}+12\frac{1}{x^2})=3$ and $\lim_{x\to \infty }(5+113\frac{1}{x}-2\frac{1}{x^2})=5$ using the fact that $\lim_{x\to \infty } \frac{1}{x}=\lim_{x\to \infty }\frac{1}{x^2}=0$
So $\displaystyle \lim_{x\to \infty }\frac{3x^2-36x+12}{5x^2+113x-2}=\frac{\lim_{x\to \infty }(3-36\frac{1}{x}+12\frac{1}{x^2})}{\lim_{x\to \infty }(5+113\frac{1}{x}-2\frac{1}{x^2})}=\frac{3}{5}$
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