Arturo O. answered • 08/23/16
Tutor
5.0
(66)
Experienced Physics Teacher for Physics Tutoring
By L'Hopital's rule, it looks like it should be
limx→0 (x + sin x)/x = limx→0 (1 + cos x)/1 = (1 + 1)/1 = 2
Fiona L.
yes please
Report
08/23/16
Arturo O.
Fiona,
This problem is an example of an indeterminate form. Both the numerator and denominator approach 0, so it is not obvious what the limiit should be. L'Hopital's rule says that the limit of a ratio of functions is the same as the limit of the ratio of the derivatives of the functions. So differentiate both the numerator and denominator, and then try evaluating the limit.
In this problem,
(x + sin x)' = 1 + cos x
x' = 1
So you evaluate the limit as x→0 of the ratio of (1 + cos x)/1. Notice that after the differentiation, the remaining form is not indeterminate, and you can evaluate the limit. Sometimes use of L'Hopital's rule produces a second indeterminate form. You can then try applying the rule successively until you end up with a form that is not indeterminate.
Report
08/23/16
Arturo O.
08/23/16