Calculus of a Single Variable, Chapter 8, 8.7, Section 8.7, Problem 38

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 .