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?

For functions such as these, the limit tends toward the ratio of the coefficients, therefore, the answer would be $\frac{3}{5}$. Below is a full calculation. 
$$\frac{3x^2-36x+12}{5x^2+113x-2}$$
If we remove an $x^2$, we will have the following
$$\frac{x^2(3-\frac{36}{x}+\frac{12}{x^2})}{x^2(5+\frac{113}{x}-\frac{2}{x^2})}$$
Now, the $x^2$'s cancel out, leaving us with the following
$$\frac{(3-\frac{36}{x}+\frac{12}{x^2})}{(5+\frac{113}{x}-\frac{2}{x^2})}$$
Now we can take the limit of each of the components seperately. Therefore, the $\lim$ of $\frac{36}{x}$ and $\frac{113}{x}$ as $x \rightarrow 0$, will be $0$. Similarly, the limit of $\frac{2}{x^2}$ and $\frac{12}{x^2}$ will be zero.
Now we are left with the limits of $3$ in the numerator and $5$ in the denominator. The limit of a constant is constant. Thus, your answer is $\frac{3}{5}$ 

In the future, if you have questions such as these or questions where the denominator has a higher power or the numerator has a higher power. You can use the following rules: 


*

*If the exponent of the highest term in the numerator matches the
exponent of the highest term in the denominator, the limit (at both
$\infty$ and $-\infty$) is the ratio of the coefficients of the highest
terms.
If the numerator has the highest term, then the fraction is called "top-heavy". If, when you divide the numerator by the
denominator the resulting exponent on the variable is even, then the
limit (at both $\infty$ and $-\infty$) is $\infty$. If it is odd, then the
limit at $\infty$ is $\infty$, and the limit at $-\infty$ is $-\infty$.
If the denominator has the highest term, then the fraction is called "bottom-heavy" and the limit (at both $\infty$ and $-\infty$) is
zero. 
Source: Wikibooks Calculus
