What is the limit of function (sin7x+sin8x)/7x, if x approaches to 0? I should not use l'Hopital rule.

First, we'll verify if we'll get an indetermination by
substituting x by the value of the accumulation
point.

lim (sin7x+sin8x)/7x  = lim (sin0 +sin
0)/7*0

We know that sin 0 =
0

lim (sin7x+sin8x)/7x = (0 - 0)/0 =
0/0

Since x approaches to 0, we'll create remarcable
limits:

lim (sin x)/x = 1, if x approaches to
0.

lim (sin7x+sin8x)/7x = lim (sin 7x)/7x + lim (sin
8x)/7x

lim (sin7x+sin8x)/7x = 1 + (1/7)* lim 8*(sin
8x)/8x

lim (sin7x+sin8x)/7x = 1 + (8/7)* lim (sin
8x)/8x

lim (sin7x+sin8x)/7x = 1 + (8/7)*
1

lim (sin7x+sin8x)/7x = 1 +
8/7

lim (sin7x+sin8x)/7x =
15/7

The limit of the given function, for x
approaches to 0, is:  lim (sin7x+sin8x)/7x =
15/7.