What is $\lim_{n\rightarrow\infty}\frac{(n+1)^3}{(n+2)n^2}$? Having some trouble remembering limit calculations.
What is limit $$\lim_{n\rightarrow\infty}\frac{(n+1)^3}{(n+2)n^2}\ \text{?}$$
 A: Notice, $$\lim_{n\to \infty}\frac{(n+1)^3}{(n+2)n^2}$$
$$=\lim_{n\to \infty}\frac{n^3\left(1+\frac{1}{n}\right)^3}{n\left(1+\frac{2}{n}\right)n^2}$$
$$=\lim_{n\to \infty}\frac{n^3\left(1+\frac{1}{n}\right)^3}{n^3\left(1+\frac{2}{n}\right)}$$
$$=\lim_{n\to \infty}\frac{\left(1+\frac{1}{n}\right)^3}{\left(1+\frac{2}{n}\right)}$$
$$=\frac{\left(1+0\right)^3}{\left(1+0\right)}=\color{red}{1}$$
A: The idea is that only the highest powers matter in such cases. 
The proof is done by dividing both nominator and denominator by $n^3$ and seeing that most of the things like $3\cdot\frac{1}{n}$ or $\frac{1}{n^2}$  tend to $0$.
In general, divide by $n$ in highest power that exists in the denominator.
For example, to calculate the limit of $\frac{n^2+2n+3}{3n+5}$ divide the nominator and the denominator by $n$. To calculate the limit of $\frac{n^2+2n+3}{3n^2+5}$, divide both the nominator and the denominator by $n^2$.
A: All this can easily be solved using the notion of equivalent functions.sequences. 
As $n$ tends to $\pm \infty$, a polynomial function is equivalent to its highest degree term, and a rational function (quotient of two polynomial functions) is equivalent to the ratio of the highest degree terms of numerator and denominator.
Here is how it goes here:
$$\frac{(n+1)^3}{(n+2)n^2}\sim_\infty\frac{n^3}{n\cdot n^2}=1,$$
hence the limit is equal to $1$.
