How do you find f'(x) using the limit definition given $f (x) = \frac{x}{x + 4}$ $f(x) = x/(x+4)$ ?

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Answer 1 · 2016-10-30T19:47:11

Please see the explanation

The limit definition is:

$lim_(hto0)(f(x + h) - f(x))/h$

We compute $f(x + h)$ :

$f(x + h) = (x + h)/(x + h + 4)$

Substitute $f(x + h) and f(x)$ into the definition:

$lim_(hto0)((x + h)/(x + h + 4) - (x)/(x + 4))/h$

Multiply the expression by 1 in the form of $((x + h + 4)(x + 4))/((x + h + 4)(x + 4))$

$lim_(hto0)((x + h)/(x + h + 4) - (x)/(x + 4))/h((x + h + 4)(x + 4))/((x + h + 4)(x + 4)) =$

$lim_(hto0)(((x + h)(x + h + 4)(x + 4))/(x + h + 4) - (x(x + h + 4)(x + 4))/(x + 4))/(h(x + h + 4)(x + 4)) =$

$lim_(hto0)((x + h)(x + 4) - (x(x + h + 4)))/(h(x + h + 4)(x + 4)) =$

$lim_(hto0)((x^2 + 4x + hx + 4h) - (x^2 + hx + 4x))/(h(x + h + 4)(x + 4)) =$

$lim_(hto0)(4h)/(h(x + h + 4)(x + 4)) =$

$lim_(hto0)4/((x + h + 4)(x + 4))$

Now it is ok to let h become 0:

$4/((x + 4)(x + 4)) =$

$4/(x + 4)^2$

How do you find f'(x) using the limit definition given f(x) = x/(x+4)f(x)=xx+4 f(x) = x/(x+4)?