Limits and derivatives

Question

lim    cos ax-cos bx .  EVALUATE.

x→0     cos cx-1              

Answer 1

Plugging in x = 0 gives 0/0, an indeterminate form, so use L'Hopital's rule: lim (cos ax - cos bx)/(cos cx - 1) = lim (-a sin ax + b sin bx)/(-c sin cx), which would still be 0/0, so use it again: = lim (-a^2 cos ax + b^2 cos bx)/(-c^2 cos cx).