Use algebra to evaluate the limit. lim h->0 ((4/(1+h)^2)-4)/h

Question

We're supposed to use (f(x+h)-f(x))/h but I'm totally lost

Answer 1

You can also do this problem without using messy algebra.

Recall that f'(x) = lim_h→0 [(f(x+h) - f(x)) / h]

So, if we let f(x) = 4 / (1+x)² and let x = 0 (so f(0) = 4), then the given limit is equal to f'(0).

f(x) = 4(1+x)^-2 So, f'(x) = -8(1+x)^-3 = -8 / (1+x)³

Therefore, f'(0) = -8

Answer 2

lim h->0 ((4/(1+h)^2)-4)/h

lim h->0 ((4-4(1+h)²)/(1+h)²)•(1/h)

Expand (1+h)² and perform the multiplication:

lim h->0 (4-4(1+2h+h² ))/(h(1+h)²)

Distribute -4 on the numerator:

lim h->0 ((4-4-8h-4h² )/(h(1+h)²)

lim h->0 ((-8h-4h² )/(h(1+h)²)

Factor out -4h

lim h->0 (-4h(2+h ))/(h(1+h)²)

Cancel h

lim h->0 -4(2+h )/(1+h)²

Now plugin 0 for h:

-4(2+0)/(1+0)² = -8 final answer