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

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### 1 Answer

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.**