Find $\lim_{n\rightarrow \infty}\int_0^1 f_n(x) dx$

Question

Let $f_n:[0, 1] \rightarrow \mathbb{R}$ be defined by $f_n(x)=\dfrac{n+x^3 \cos x}{n e^x + x^5 \sin x}, n \geq 1$. Find $\lim_{n\rightarrow \infty}\int_0^1 f_n(x) dx$

My answer is $1-\dfrac{1}{e}.$ Please see it, right or wrong.

Answer 1

For $x\in[0,1]$

$$\lim_{n\to+\infty}f_n(x)=e^{-x}$$

let $f(x)=e^{-x}$.

if the convergence is uniforme then

$$\lim_{n\to+\infty}\int_0^1f_n(x)dx=\int_0^1f(x)dx$$

but $$|f_n(x)-f(x)|=\frac{|x^3\cos(x)-x^5\sin(x)e^{-x}|}{ne^x+x^5\sin(x)}$$

$$\le \frac{2}{ne^x+x^5\sin(x)}\le \frac 2n$$

the convergence is now uniforme and the limit is

$$\int_0^1e^{-x}dx=\Bigl[-e^{-x}\Bigr]_0^1$$

$$=1-\frac 1e$$

So, your answer is right.