Prove that the limit of the following sequence $\frac{n^3 + 2}{n^2 + 3}$ diverges to $+ \infty$.

Question

The question is:

Prove that the limit of the following sequence $\frac{n^3 + 2}{n^2 + 3}$ diverges to $+ \infty$.

My choice to $N$ was the floor of $M/2$ plus 1, am I right?

where I found a sequence larger than the given sequence which is $2n$ and where in my definition I have the assumption that $n \geq N$.

Answer 1

You wish to show that $f(n) \geq M$ for all $n \geq N$.

What you have doesn’t work. For example taking $M=3$ gives $N=2$. But $f(2)=\frac{10}{7} < 3$.

Answer 2

$$\dfrac{n^3 + 2}{n^2 + 3}>\frac{n^3 }{n^2 + n^2}=\dfrac12n>N$$ so it is sufficient to let $M=2N$.