limit of sin(x)/x as x approaches infinity

Question

I believe it equls 1, but how?  I tried L'Hopital's Rule and still can not get just a Cos(x) as x approaches infinity

Answer 1

If the form is indeterminate, then apply L'Hopital's rule.  However, in this problem, the form is not indeterminate, because the denominator goes to infinity while the numerator remains finite, so the form approaches zero.

 

L'Hopital's rule works fine for a problem like:

 

Limit as x → 0 of sin(x)/x.

 

In that case, the form is indeterminate, and L'Hopital's rule gives 1 for the limit.  But your stated problem has x → ∞.  As the tutors explained, in the stated problem you can reason your way to the conclusion that the limit is zero.

Answer 2

The sine function oscillates between -1 and 1 for all x. As x approaches infinity, that means the denominator will also approach infinity. The limit of this function will be zero as x approaches infinity. 

Answer 3

 One uses the squeeze rule to show this, squeezing the function in absolute value between 0 and 1/x. Since 1/x approaches 0, so does sin x / x.