Use the rules f limits to find the answer to m at equation below?

Question

use the rules of limits to find
m= lim f(a-h)- f(a)/ h as h--> 0
for f(x)= x^(3/2) and a= 4. Explain how your answer is related to either the graph of f, or the function f.

Answer 1

I believe thatthe limit is actually +3

as this is the definaition of the derivative

f'(4) = (3/2)*4^(1/2) = 3/2*2 = 3.

*I'm sure it was only a slight error in computation.

Answer 2

lim [ (4 - h)^(3/2) - (4)^(3/2) ] / h
h -> 0

First off, 4^(3/2) is equal to 8, so we have

lim [ (4 - h)^(3/2) - 8 ] / h
h -> 0

Multiply numerator and denominator by the conjugate of the numerator. That is, [(4 - h)^(3/2) + 8]. This will lead to a difference of squares on the top.

lim [ ( (4 - h)^(3/2) )^2 - 64 ] / h[(4 - h)^(3/2) + 8]
h -> 0


lim [ (4 - h)^3 - 64 ] / h[(4 - h)^(3/2) + 8]
h -> 0

What we have now is a difference of cubes on the top. We know how to factor difference of cubes.

lim (4 - h - 4) [ (4 - h)^2 + 4(4 - h) + 16] / h[(4 - h)^(3/2) + 8]
h -> 0

Look at the first set of brackets and how easily it simplifies to -h.

lim (-h) [ (4 - h)^2 + 4(4 - h) + 16] / h[(4 - h)^(3/2) + 8]
h -> 0

And look how the h cancels on the top and bottom.

lim (-1) [ (4 - h)^2 + 4(4 - h) + 16] / [(4 - h)^(3/2) + 8]
h -> 0

At this point, we can now directly plug in h = 0. This gives us

(-1) [ (4 - 0)^2 + 4(4 - 0) + 16] / [(4 - 0)^(3/2) + 8]

(-1) [4^2 + 4(4) + 16] / [4^(3/2) + 8]
(-1) [16 + 16 + 16] / [8 + 8]
(-1) [48] / [16]
(-1) (3)

= -3