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

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Answer 1 · 2014-08-18T02:42:41

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 $x/x$ , and the second term by $4/4$ :

$= lim_(x->-4) ((x/(4x)) + (4/(4x)))/(x+4)$

Now, we can combine the terms:

$= lim_(x->-4) ((x+4)/(4x))/(x+4)$

Simplifying gives us:

$= lim_(x->-4) (x+4)/(4x(x+4))$

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

$= lim_(x->-4) 1/(4x)$

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

$= 1/(4*(-4)) = -1/16$

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