# Evaluate $\lim_{n \rightarrow \infty} \int_{[0,1]} x\cos(\frac{x}{n})$

Evaluate $$\lim_{n \rightarrow \infty} \int_{[0,1]} x\cos(\frac{x}{n})$$.

So, I'm pretty sure I'm supposed to use the Lebesgue dominated convergence theorem? I could use the Vitali convergence theorem, since the domain is finite. Struggling with this one, would appreciate some help, thanks!

• the answer is $\frac{1}{2}$ right...? – Saketh Malyala Dec 8 '19 at 0:39
• Hint: Use Lebesgue and that the integrand goes to $x$ as $n\to\infty$ – Maximilian Janisch Dec 8 '19 at 0:41

$$|x \cos(\frac x n) | \leq x$$ and $$x$$ is integrable on $$[0,1]$$. Hence the limit is $$\int_0^{1}x dx=\frac 1 2$$.
• Okay, so we have a dominating function, but don't we also need the pointwise limit of $x cos(\frac{x}{n})$? Doesn't this function have no limit? – MSV Dec 8 '19 at 0:43