Lebesgue integration: Showing $\lim\limits_{\lambda \rightarrow \infty} \int_{0}^{\infty} e^{-x}\cos(x)\arctan(\lambda x) \ dx = \frac{\pi}{4}$

Question

I am trying to show that:

$\displaystyle\lim_{\lambda \rightarrow \infty} \int_{0}^{\infty} e^{-x}\cos(x)\arctan(\lambda x) \ dx = \dfrac{\pi}{4}$

I've tried using MCT/DCT but haven't found a suitable expression for the above to use either. I'm also somewhat stuck because assuming the integral and limit can be swapped I don't know exactly how to evaluate $\displaystyle\lim_{\lambda \rightarrow \infty} \arctan(\lambda x)$

Thanks

Answer 1

We have $$\lim_{\lambda \to\infty} \arctan(\lambda x)=\left\{\begin{array}\\\frac\pi2&\text{if}&x>0\\0&\text{if}&x=0\end{array}\right.\;\;=\frac\pi2\;\text{a.e.}$$ and $$\left|e^{-x}\cos(x)\arctan(\lambda x)\right|\le\frac\pi2 e^{-x}\in L^1([0,\infty))$$ so by the dominated convergence theorem we have $$\displaystyle\lim_{\lambda \rightarrow \infty} \int_{0}^{\infty} e^{-x}\cos(x)\arctan(\lambda x) \ dx =\frac\pi2\int_{0}^{\infty} e^{-x}\cos(x) \ dx=\Re \frac\pi2\int_{0}^{\infty} e^{(-1+i)x} \ dx = \dfrac{\pi}{4}$$