`lim_(x->1) lnx/sin(pix)` Evaluate the limit, using L’Hôpital’s Rule if necessary.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`lim_(x->1) (ln(x))/(sin(pix))`

To solve, plug-in x = 1.

`lim_(x->1) (ln(x))/(sin(pix)) = (ln(1))/(sin(pi*1)) = 0/0`

Since the result is indeterminate, to find the limit of the function as x approaches 1, apply L'Hopital's Rule. So, take the derivative of the numerator and the denominator.

`lim_(x->1) (ln(x))/(sin(pix)) =lim_(x->1) ((ln(x))')/((sin(pix))') = lim_(x->1) (1/x)/(pi cos(pix)) = lim_(x->1) 1/(pix cos(pix))`

And, plug-in x = 1.

`= 1/(pi*1*cos(pi*1))=1/(pi*cos(pi)) = 1/(pi*(-1)) = -1/(pi)`


Therefore,  `lim_(x->1) (ln(x))/(sin(pix))=-1/pi` .

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial