Determine the limit of a function based on a given function I received this question on my past midterm and the professor did not offer a solution and I am perplexed as to what the answer is. 
Suppose f is a function such that
$${\lim_{x\rightarrow \infty }} [f(x) + 2x^{2}] = 1$$
Determine, if possible,
$$\lim_{x\rightarrow \infty} {\frac{f(x)}{x^{2}+1}}$$

Write: $\frac{f(x)}{x^2+1} = \frac{f(x)+2x^2}{x^2+1} - \frac{2x^2}{x^2+1}$.
We know that $\lim_{x \to \infty} \frac 1{x^2+1} = 0$. Hence, by product of limits formula:
$$
\lim_{x \to \infty} \frac{f(x)+2x^2}{x^2+1} = \lim_{x \to \infty} [f(x)+2x^2] \lim_{x \to \infty} \frac 1{x^2+1} = 1 \times 0 = 0
$$
Note that $\frac{2x^2}{x^2+1} = 2 - \frac{2}{x^2+1}$, so the limit of this quantity as $x \to \infty$ is $2$.
Hence, the limit of $\frac{f(x)}{x^2+1}$ exists, and equals $0 - 2 = -2$.
EXAMPLE : $f(x) = -2x^2 + 1$, then $f$ satisfies the conditions given, and $\frac{f(x)}{x^2+1}= \frac{-2x^2 - 2 + 1}{x^2+1} = -2 +\frac{1}{x^2+1}$, whence the limit is $-2$.
