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Story: Watch Online Slumdog Millionaire 2008 Dual Audio 480p BluRay x264 400MB HDRip.Q:
Prove continuity of expression
I have to prove that $$f(x)=\int_0^1y^3\sin{(xy)}dx$$ is continuous at $x=0$.
I take $x_n\rightarrow0$ and $x_n
eq0$
And now I need to compute the limit of the expression $$\lim_{x_n\rightarrow0}\int_0^1(x_n-x)^3\sin{(x_n-x)}dx$$
As I understand I have to use $$\sin{(x_n-x)}=\sin{x_n}\cos{x}-\cos{x_n}\sin{x}$$
But what is missing here?
A:
Integrate,
$$\begin{align}
\int_0^1(x_n-x)^3\sin{(x_n-x)}dx
&=\left[\left(x_n-x\right)^4\sin{(x_n-x)}\right]_0^1-
\int_0^14\left(x_n-x\right)^3\cos{(x_n-x)}\sin{(x_n-x)}dx\\
&=4\left(x_n-x\
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