Hello there. To solve this question, we have to remember some properties about limits.
Given the following limit:
[tex]\begin{gathered} \lim_{x\to\infty}\dfrac{\cos(5x)}{x} \\ \end{gathered}[/tex]
We want to determine its value.
For this, we'll use the "sandwich" theorem, that is also called as the squeeze theorem.
Notice that
[tex]-1\leq\cos(5x)\leq1[/tex]
Hence dividing both sides of the equation by a factor of x, we'll get
[tex]-\dfrac{1}{x}\leq\dfrac{\cos(5x)}{x}\leq\dfrac{1}{x}[/tex]
Taking the limit as x goes to infinity (and of course this works for x very large), it wouldn't work if we were to determine the value at 0.
[tex]\lim_{x\to\infty}-\dfrac{1}{x}\leq\lim_{x\to\infty}\dfrac{\cos(5x)}{x}\leq\lim_{x\to\infty}\dfrac{1}{x}[/tex]
The left and right hand side limits are equal to zero, hence
[tex]0\leq\lim_{x\to\infty}\dfrac{\cos(5x)}{x}\leq0[/tex]
And this is precisely the value of this limit:
[tex]\lim_{x\to\infty}\dfrac{\cos(5x)}{x}=0[/tex]
