Question

Hi I was wondering if you could confirm why the limit is 0

Answer 1

Hello there. To solve this question, we have to remember some properties about limits.

Given the following limit:

`\begin{gathered} \lim_(x\to\infty)(\cos(5x))/(x) \\ \end{gathered}`

We want to determine its value.

For this, we'll use the "sandwich" theorem, that is also called as the squeeze theorem.

Notice that

`-1\leq\cos(5x)\leq1`

Hence dividing both sides of the equation by a factor of x, we'll get

`-(1)/(x)\leq(\cos(5x))/(x)\leq(1)/(x)`

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.

`\lim_(x\to\infty)-(1)/(x)\leq\lim_(x\to\infty)(\cos(5x))/(x)\leq\lim_(x\to\infty)(1)/(x)`

The left and right hand side limits are equal to zero, hence

`0\leq\lim_(x\to\infty)(\cos(5x))/(x)\leq0`

And this is precisely the value of this limit:

`\lim_(x\to\infty)(\cos(5x))/(x)=0`