$\lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} \sin(\alpha x)\cos(\beta x) \,dx$

Question

I have to compute this limit:

Suppose that $|\alpha| \neq |\beta |$, compute the following limit: \begin{align} \lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} \sin(\alpha x)\cos(\beta x) \, dx \end{align}

What I've done is this: \begin{align} & \int_0^T \sin(\alpha x)\cos(\beta x) \, dx \\[8pt] = {} & \frac{1}{2(\alpha-\beta)} + \frac{1}{2(\alpha+\beta)}-\frac{\cos((\alpha-\beta)T)}{2(\alpha-\beta)}-\frac{\cos((\alpha+\beta)T)}{2(\alpha+\beta)} \end{align}

Then,

\begin{align} \lim_{T \rightarrow \infty}\frac{1}{T}\int_0^T \sin(\alpha x) \cos(\beta x) \,dx=\lim_{T \rightarrow \infty}\frac{1}{T}\left [ \frac{1}{2(\alpha-\beta)}+\frac{1}{2(\alpha+\beta)}-\frac{\cos((\alpha-\beta)T)}{2(\alpha-\beta)}-\frac{\cos((\alpha+\beta)T)}{2(\alpha+\beta)} \right ] \end{align}

But I'm not sure how can I conclude. What I think that I can do is this:

\begin{align} \underbrace{\lim_{T \rightarrow \infty}\frac{1}{T}}_{=0}\cdot \underbrace{\lim_{T \rightarrow \infty}\left [ \frac{1}{2(\alpha-\beta)}+\frac{1}{2(\alpha+\beta)}-\frac{\cos((\alpha-\beta)T)}{2(\alpha-\beta)}-\frac{\cos((\alpha+\beta)T)}{2(\alpha+\beta)} \right ]}_{\text{I don't know how to compute it}} \end{align}

So intuitively I think this must be zero. How can I conclude? Is there any other way to solve it? I really appreciate your help.

Answer 1

Your last step doesn't work because $\lim_{T \rightarrow \infty}\left [ \frac{1}{2(\alpha-\beta)}+\frac{1}{2(\alpha+\beta)}-\frac{\cos((\alpha-\beta)T)}{2(\alpha-\beta)}-\frac{\cos((\alpha+\beta)T)}{2(\alpha+\beta)} \right ]$ does not exist. It doesn't exist becuase the cosines oscillate rather than approaching a limit.

But you can compute $$ \lim_{T \rightarrow \infty}\frac{1}{T}\left [ \frac{1}{2(\alpha-\beta)}+\frac{1}{2(\alpha+\beta)}-\frac{\cos((\alpha-\beta)T)}{2(\alpha-\beta)}-\frac{\cos((\alpha+\beta)T)}{2(\alpha+\beta)} \right ] $$ by squeezing. You have $$ \lim_{T\to\infty} \frac 1 T \Big[\cdots\cdots\cdots \Big] $$ where the expression in the $\displaystyle\Big[\cdots\text{ large square brackets }\cdots\Big]$ remains bounded, i.e. remains between two identifiable bounds, as $T$ changes. Thus you seek the limit of an expression that is squeezed between two things that approach $0.$