# Find the limit as x approaches infinity of #xsin(1/x)#?

##### 3 Answers

By l'Hopital's Rule, we can find

Let us look at some details.

by rewriting a little bit,

by l'Ho[ital's Rule,

by cancelling out

Instead of l'Hopital's Rule, one can use the fundamental trigonometric limit:

The limit you are interested in can be written:

Now, as

With

Although it is NOT needed, here's the graph of the function:

graph{y = x sin(1/x) [-5.55, 5.55, -2.775, 2.774]}

When you substitute in infinity,

We still have options though. We now can fall back on L'Hopital's Rule which basically says to take the derivative of the numerator and denominator independently. Do not use the quotient rule.

We need to rewrite this function so that is produces an indeterminate in the form

**Applying L'Hopital**

**Simplify the previous step**