Evaluate the limit.

Question

lim x → 1 ln x/sin 5πx

Answer 1

lim [ ln (x)/(sin (5πx)) ] as x → 1

Use L'hopital's rule. Differentiate numerator and denominator separately:
lim [ (1/x) / 5π(cos(5πx)) ] as x → 1

Substitute x = 1:
= (1/1) / 5π(cos(5π(1)))
= 1 / 5π(-1)
= -1 / 5π