<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] $$\lim_{x\to\infty}\frac{ 3x^2 - 36x+12}{5x^2+113x-2}$$
To solve this you have  two ways, one very simple is  to apply L'hopitals' rule two times and get 
$$\lim_{x\to\infty}\frac{ 3x^2 - 36x+12}{5x^2+113x-2}=
\lim_{x\to\infty}\frac{ 6x- 36}{10x+113}=\lim_{x\to \infty}=6/10=3/5$$
or you ca divide by $x^2$ as follow
$$\lim_{x\to\infty}\frac{ 3x^2 - 36x+12}{5x^2+113x-2}=
\lim_{x\to\infty}\frac{ \frac{3x^2 - 36x+12}{x^2}}{\frac{5x^2+113x-2}{x^2}}
= \frac{\displaystyle\lim_{x\to\infty} 3 - 36/x+12/x^2}{\displaystyle \lim_{x\to\infty}5+113/x-2/x^2}= \frac{3}{5}$$
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