How do you find the limit lim_(x->-4)((1/4)+(1/x))/(4+x) ?

Aug 18, 2014

This is a type of problem where the function inside the limit just needs to be simplified until the answer is apparent.

We will simplify the numerator by multiplying the first term by $\frac{x}{x}$, and the second term by $\frac{4}{4}$:

$= {\lim}_{x \to - 4} \frac{\left(\frac{x}{4 x}\right) + \left(\frac{4}{4 x}\right)}{x + 4}$

Now, we can combine the terms:

$= {\lim}_{x \to - 4} \frac{\frac{x + 4}{4 x}}{x + 4}$

Simplifying gives us:

$= {\lim}_{x \to - 4} \frac{x + 4}{4 x \left(x + 4\right)}$

The $x + 4$ term will cancel, leaving us with:

$= {\lim}_{x \to - 4} \frac{1}{4 x}$

The solution is now easily found by substituting $x = - 4$:

$= \frac{1}{4 \cdot \left(- 4\right)} = - \frac{1}{16}$