# Question e4230

Apr 2, 2016

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

#### Explanation:

First of all:

${\lim}_{x \to 0} \frac{5 x + \sin x}{x} = {\lim}_{x \to 0} \left[\frac{5 x}{x} + \frac{\sin x}{x}\right] =$

$= {\lim}_{x \to 0} \left[5 + \sin \frac{x}{x}\right] = 5 + {\lim}_{x \to 0} \sin \frac{x}{x}$

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:

${\lim}_{x \to 0} \sin \frac{x}{x} = {\lim}_{x \to 0} \frac{x}{x} = {\lim}_{x \to 0} 1 = 1$

Hence:

${\lim}_{x \to 0} \frac{5 x + \sin x}{x} = 5 + 1 = 6$