lim as X approches 0 (sin9x)/(sin4x); evalute the limit.?

Answer 1

As x --> 0, lim sin(9x)/sin(4x) = lim sin(9x)/(9x) * 9x * (4x)/sin(4x) * (1/4x) = lim sin(9x)/(9x) * (4x)/sin(4x) * 9/4

We know that lim x --> 0 sin(x) /x =1. Therefore

lim x --> 0 sin(9x)/(9x) = 1 and lim x --> 0 (4x)/sin(4x) =1

By the properties of limits, it follows that

lim x --> 0 sin(9x)/sin(4x) = 1 * 1 * 9/4 = 9/4

By the same reasoning, it follows that, if a and b<>0 are real numbers, then lim x->0 sin(ax)/sin(bx) = a/b