How to apply squeeze theorem to this limit. I'm trying to solve $$\int_0^∞ e^{-x} \cos(x)\,dx$$
It is not hard to find that $$\int e^{-x} \cos(x)=\frac{1}{2}(e^{-x} \sin(x)-e^{-x} \cos(x))+C$$
From all this follows that
$$\lim_{t\to\infty}\int_0^te^{-x} \cos(x) \, dx = \frac{1}{2}\lim_{t\to\infty}(e^{-t} \sin(t) - e^{-t} \cos(t))+\frac{1}{2}$$
Notice that I have simplified already a lot the expression we are taking the limit of.
I have not been able to find this limit; a collegue student told me that I had to use the squeeze theorem, but I do not find how nor where. Any guides on how the theorem can help with this limit?
 A: The values of trigonometric functions oscillate between $1$ and $-1$.
Your upper bound is:
$$
\lim_{t \to \infty} \int_{0}^{t} e^{-x}\cos(x)\,dx =
\frac{1}{2} \lim_{t \to \infty}(e^{-t}(1) - e^{-t}(-1)) + \frac{1}{2}.
$$
We don’t know what value our trigonometric functions take at infinity but they can’t take any larger value clearly. So we set $\cos$ to $-1$ and $\sin$ to $1$ as it will give $\frac{1}{2}(2e^{-t})+\frac{1}{2}$.
And our lower bound is:
$$
\lim_{t \to \infty} \int_{0}^{t} e^{-x}\cos(x)\,dx =
\frac{1}{2} \lim_{t \to \infty}(e^{-t}(-1) - e^{-t}(1)) + \frac{1}{2}.
$$
Which gives us $\frac{1}{2}(-2e^{-t}) + \frac{1}{2}$.
It’s clear that $e^{-t}$ goes to zero and so our limit is ‘squeezed’ to $\frac{1}{2}$ from both sides.
A: It is easiest to compute this integral using the identity $\cos x = \frac12(e^{ix}+e^{-ix})$. From this we have
\begin{align}
\int_0^\infty e^{-x}\cos x\ \mathsf dx &= \frac12\int_0^\infty e^{-x}(e^{ix}+e^{-ix})\ \mathsf dx\\
&= \frac12\int_0^\infty \left((e^{(i-1)x}+e^{-(i+1)x}\right)\ \mathsf dx\\
&=\frac12\left[\frac1{i-1}e^{(i-1)x} - \frac1{i+1}e^{-(i+1)x} \right]_0^\infty\\
&= \frac12\left(\frac1{i+1} - \frac1{i-1} \right)\\
&=\frac12\left(\frac1{1+i} + \frac1{1-i}\right)\\
&=\frac12\left(\frac{1-i + 1 + i}{(1+i)(1-i)} \right)\\
&= \frac12\left(\frac2{1-i^2} \right)\\
&= \frac12.
\end{align}
A: $$\because\sin x-\cos x=\sqrt 2(\cos\frac{\pi}{4}\sin x-\sin\frac{\pi}{4}\cos x)=\sqrt 2\sin (x-\frac{\pi}{4})$$
$$\therefore\lim_{t\rightarrow +\infty}|e^{-t}\sin t-e^{-t}\cos t|= \lim_{t\rightarrow +\infty}|\sqrt 2e^{-t}\sin (t-\frac{\pi}{4})|\leqslant \lim_{t\rightarrow +\infty}|\sqrt 2e^{-t}|=0$$
$$\therefore\lim_{t\rightarrow +\infty}(e^{-t}\sin t-e^{-t}\cos t)=0$$
$$\therefore\frac{1}{2}\lim_{t\rightarrow +\infty}(e^{-t}\sin t-e^{-t}\cos t)+\frac{1}{2}=\frac{1}{2}$$
A: \begin{align}
& \lim_{t\to\infty} \left( e^{-t} \sin t - e^{-t} \cos t \right) \\[8pt]
= {} & \left( \lim_{t\to\infty} e^{-t}\sin t \right)  - \left( \lim_{t\to\infty} e^{-t} \cos t \right) \\
& \text{provided these last two limits exist} \\
& \text{(and are finite).}
\end{align}
Then, since $-1 \le \sin t \le 1,$ we have
$$
-e^{-t} \le e^{-t}\sin t \le e^{-t}.
$$
The first and third expressions above both approach $0$ as $t\to\infty.$
(And the one involving the cosine is done the same way.)
