# Concerning the limit of the numerator of a rational function

I have a basic question about the limits of rational sequences:

Suppose we know that a sequence $$(a_{n})$$ converges to $$0$$ and that each $$a_n$$ is defined by $$a_n := \frac{b_n}{c_n}$$ for two sequences $$(b_{n})$$ and $$(c_{n})$$. We further know that $$(c_{n})$$ goes to infinity, i.e

$$\lim_{n \rightarrow \infty} c_n = \infty.$$

Can we deduce from this that $$\lim_{n \rightarrow \infty} \ b_n = 0$$ and if yes how?

Edit: I have an additional question: Are we at least able to deduce that $$b_{(n)}$$ converges?

No.

What about $$\left( a_n \right)=\left( \frac{1}{n}\right)$$

where $$a_n=\frac{1}{n}$$, $$b_n=1$$ and $$c_n=n$$

Again No.

What about $$\left( a_n \right)=\left( \frac{n}{n^2}\right)$$

where $$a_n=\frac{n}{n^2}$$, $$b_n=n$$ and $$c_n=n^2$$

No, that’s not true. For instance, for $$b_n = n$$ and $$c_n = n^2$$, we have

$$\lim_{n \to \infty}\frac{b_n}{c_n} = 0$$

while both sequences tend to $$\infty$$. More generally, when both $$b_n$$ and $$c_n$$ are polynomials with $$c_n$$ having the higher degree, the limit tends to $$0$$, even though both polynomials may tend to $$\infty$$.

Edit: Also no, the example above once again shows this. Both polynomials clearly diverge to $$\infty$$, but since the denominator has a greater degree, it diverges faster. Hence, the limit tends to $$0$$.