what is the limit (x->0) sin(x)/x

Question

limits question.

Answer 1

This is a classic example where using the L'Hôpital rule will get you an answer. Since both sin(x) and x approach 0 as x->0, you have the case "0 / 0" in the limit.

 

The L'Hôpital rule says that in such cases ("0/0", "inf/inf"), the limit can be obtained by taking the derivative of both numerator and denominator.

 

The derivative of sin(x) is cos(x). The derivative of x is simply 1. L'Hôpital says that the limit of sin(x)/x (x->0) is the same as the limit of cos(x)/1 as x->0.

 

The limit of cos(x) (x->0) is cos(0) = 1. Therefore the limit of sin(x)/x (x->0) is 1.

Answer 2

This is a special case and the limit would always be equal to 1