Which function is "larger" when variable aims to infinity?

Question

Suppose I have $2$ different functions:

1) $n^2$

2) $(n+1)^2$

now doing this I get: $\lim_{n \to \infty}\frac{(n+1)^{2}}{n^{2}}$ which equals to 1. I "know" that when $\infty$ then numerator is bigger than denominator and "0" looking the opposite direction. in thi case I get 1. What does it mean?

Answer 1

It means they are essentially equivalent at infinity. They have the same order of growth. As $n$ becomes larger, their quotient becomes closer and closer to 1.

Answer 2

Note that

$$(n+1)^2>n^2$$

and indeed

$$(n+1)^2-n^2=2n+1 \to \infty$$

therefore $(n+1)^2$ is larger than $n^2$.

The fact that the ratio

$$\frac{(n+1)^2}{n^2}\to1$$

tell us that both the sequecences tends to infinity with the same "speed" or with the same rate of growth.