Question

How do you find the limit of `(1/(h+2)^2 - 1/4) / h` as h approaches 0?

Answer 1

We need first to manipulate the expression to put it in a more convenient form

Explanation:

Let's work on the expression

`(1/(h+2)^2 -1/4)/h=((4-(h+2)^2)/(4(h+2)^2)) /h=((4-(h^2+4h+4))/(4(h+2)^2)) /h=(((4-h^2-4h-4))/(4(h+2)^2)) /h=(-h^2-4h)/(4(h+2)^2 h) = (h(-h-4))/(4(h+2)^2 h) = (-h-4)/(4(h+2)^2)`

Taking now limits when `h-> 0` we have:

`lim_(h->0)(-h-4)/(4(h+2)^2) = (-4)/16=-1/4`