Prove that $\lim\limits_{n\rightarrow \infty}\int_1^3\frac{nx^{99}+5}{x^3+nx^{66}} d x$ exists and evaluate it.

Solution 1:

You are on the good way, and to complete the proof you can by triangle inequality do: $$\left|\frac{5-x^{36}}{x^3+nx^{66}}\right|\leq \frac{5+x^{36}}{x^3+nx^{66}}\leq\frac{5+3^{36}}{1^3+n1^{66}}= \frac{5+3^{36}}{1+n}\rightarrow0,\quad\forall x\in[1,3]. $$

Hence, $\forall \epsilon>0,$ we can find $N\in\mathbb{N}$ such that $\forall x\in [1,3],\quad\forall n\geq N$ we have $\left|\frac{5-x^{36}}{x^3+nx^{66}}\right|\leq \epsilon. $ Now, you can conclude the uniform convergence.

Solution 2:

The integrand simplifies to $\dfrac{nx^{96}}{1+nx^{63}}+\dfrac{5}{x^3+nx^{66}}$.

The second term is less than $\dfrac{5}{n}$ on our interval, so it is harmless.

For the first term, divide. We get $x^{33}-\dfrac{x^{33}}{1+nx^{63}}$. The function $\dfrac{x^{33}}{1+nx^{63}}$ is less than $\dfrac{1}{n}$ on our interval.