Finding the limit of these functions. how do I find the limit of these functions?
limit of x approaches infinity, $\displaystyle \lim_{x \to \infty}\displaystyle \frac{x^{10}-x^3}{x^9+1}$
limit of x approaches zero, $\displaystyle \lim_{x \to 0}\displaystyle \frac{x^3-1}{x^3+1}$
I'm pretty much confused. Thanks for the help!
 A: Hints: pay attention to the fact that in both cases we do the same "trick" with the highest power of $\,x\,$ in the expression...
$$\begin{align*}\bullet&\;\;\frac{x^{10}-x^3}{x^9+1}\cdot\frac{\frac1{x^{10}}}{\frac1{x^{10}}}=\frac{1-\frac1{x^7}}{\frac1x+\frac1{x^{10}}}\\{}\\
\bullet&\;\;\frac{x^3-1}{x^3+1}\cdot\frac{\frac1{x^3}}{\frac1{x^3}}=\frac{1-\frac1{x^3}}{1+\frac1{x^3}}\end{align*}$$
Remember: use arithmetic of limits and
$$\lim_{x\to\infty}\frac1{x^n}=0\;\;,\;\text{for}\;\;n\in\Bbb N$$
A: You just need to plug in $0$ for the second one, unless you're looking for an $\epsilon$-$\delta$ proof of it.
$$\displaystyle \lim_{x \to 0} \frac{x^3-1}{x^3+1}=-1 $$
For the first one, you can use factorize as DonAntonio did or you can use L'hopital's rule if you know what it is. 
In general, if you have a polynomial $P(x)=a_0+a_1x+\cdots + a_{n-1}x^{n-1}+a_nx^n$ and $x \to \infty$  then $P(x) \approx a_nx^n$. That means when dealing with the limits where $x \to \infty$ you can think that only the highest degree term of the polynomial matters.
