Limits to infinity, even and odd functions I have a couple of questions regarding a practice test I just made, so the subject might vary a little bit but most of it has to do with limits.


*

*$$ \lim_{x \to \infty} \dfrac{7x+3x^2}{1-x^3} $$


Apparently this is $-3$, but I've read in my book that if a rational function has a higher order polynomial in the denominator, that the limit always becomes $0$. If this is false, how are you supposed to evaluate this limit?


*$$ \lim_{x \to 1+} \dfrac{7x+3x^2}{1-x^3} $$


I just have no idea how to do this, right and left limits are very difficult to me unless I have the graph. 


*$g(x) = f(x) - f(-x)$ and $h(x) = f(x) + f(-x)$. What is the parity of $g$ and $h$. $f$ is a function with domain $\mathbb{R}$. 


How are you supposed to do this without knowing the parity of $f$?


*$$ \lim_{x \to \infty} \dfrac{1}{x-\sqrt{x^2+ax+b}} $$


$a,b \in \mathbb{R}$ with $a\neq 0$. 
I haven't worked with 2 variables yet, so I would appreciate a good tactic here.
 A: For the first problem, the limit is indeed $0$. 
For the second problem, imagine that $x$ is a tiny bit bigger than $1$. What can you say about $1-x^3$? What about $\frac{7x+3x^2}{1-x^3}$? It should be clear, but if it isn't, use your calculator with $x=1.00001$. 
For the third problem, let us do half, showing that $h$ is even.
We have 
$$h(-x)=f(-x)+f(-(-x))=f(-x)+f(x)=h(x).$$ 
For the fourth problem, multiply top and bottom by $x+\sqrt{x^2+ax+b}$.
A: Here are the steps for the first problem
$$
\lim_{x \to \infty} \frac{7x+3x^2}{1-x^3}=\lim_{x \to \infty} \frac{\frac{7x}{x^2}+\frac{3x^2}{x^2}}{\frac{1}{x^2}-\frac{x^3}{x^2}}=\lim_{x \to \infty} \frac{\frac{7}{x}+3}{\frac{1}{x^2}-x}=\frac{\frac{7}{\infty}+3}{\frac{1}{\infty}-\infty}=-\frac{3}{\infty}=0
$$
For the second problem, we have
$$
\lim_{x \to 1+} \frac{7x+3x^2}{1-x^3}
$$
Since $1 \lt x^3$ as $x\to 1$ from the right, then the denominator is negative. This expression also has a vertical asymptote at $x=1$, therefore
$$
\lim_{x \to 1+} \frac{7x+3x^2}{1-x^3}=-\infty
$$
Give the others a shot and feel free to ask questions.
