Computing the limit $\lim_{n\to\infty} (n^3x^{3/4})/(1+n^4x^2)$

Question

Find $$\lim_{n \rightarrow \infty} \frac{n^3 x^{3/4}}{ 1 + n^4 x^2}.$$

The overall goal is to find the uniform limit of a sequence of functions, or show that the sequence does not converge uniformly.

My attempt:

I know that I will treat $x$ as a constant here and I can divide both the numerator and denominator by the largest power of $n$ in the denominator, which is $n^4$. but then what? I want to find to what function the given sequence of functions converges uniformly, could anyone help me in doing so?

Answer 1

Pointwise limit is $0$. If the convergence is uniform then the expression must tend to $0$ even when you make $x$ depend on $n$.

If you put $x_n=n^{-4/5}$ the expression tends to $1$ as $n \to \infty$. This implies that the convergence is not uniform.

Proof of the fact that pointwise limit is $0$: if $x=0$ this is obvious. If $x \neq 0$ divide numerator and denominator by $n^{4}$ and take the limit.

Answer 2

As the power of $x$ in denominator is greater than nominator, they can't uniformly converge at $x=0$. If you want, you should exclude an interval around $x=0$ from domain.