Question

Question #e4230

Answer 1

Let us solve the limit by solving the limit of an addition as the addition of the limits.

Explanation:

First of all:

`lim_{x to 0} {5x + sin x}/{x} = lim_{x to 0} [{5x}/x + {sin x}/{x}] =`

`=lim_{x to 0} [5 + sin x / x] = 5 + lim_{x to 0} sin x / x`

For small values of `x`, near to 0, we can approximate:

`(sin x ~ x) leftrightarrow (lim_{x to 0} sin x = x)`

(You can check it with a calculator: for small angles, sine approximates to the angle. Note: use your calculator in radians mode).

So:

`lim_{x to 0} sin x / x = lim_{x to 0} x / x = lim_{x to 0} 1 = 1`

Hence:

`lim_{x to 0} {5x + sin x}/x = 5 + 1 = 6`