Simplifying...hard problems!!!?

Question

Hi,

I have 3 problems, I will also provide the answers, I need the work, and I have no idea on how to get started.

First one to give answers will be chosen Best, and they must be detailed and full~!

Simplify (f(x+h)-f(x)) / h

where...

a)f(x)=2x+3

b)f(x)= 1/(x+1)

c)f(x)=x^2

answers:

a)2

b)−1/ ((x + 1)(x + h + 1))

c)2x+h

Thanks for anyone who gives an Honest attempt

Answer 1

Let A = (f(x+h) - f(x))/h

For f(x) = 2x + 3, f(x+h) = 2(x+h)+3

=> A = (2x+2h+3-2x-3)/h

=> A = 2h/h = 2

for f(x) = 1/(x+1), f(x+h) = 1/(x+h+1)

=> A = (1/(x+h+1) - 1/(x+1))/h

=> A = ((x+1-x-h-1)/(x+h+1)(x+1))/h

=> A = (-h)/h*1/(x+h+1)(x+1)

=> A = -1/(x+h+1)(x+1)

For f(x) = x^2, f(x+h) = (x+h)^2 = x^2 + 2xh + h^2

=> A = (x^2 + 2xh + h^2 -x^2)/h

=> A = (2xh+ h^2)/h = 2x +h

Answer 2

All you have to do is plug your specific equation into the general equation for a limit and simplify.

A)

2(x+h) + 3 - 2x - 3 / h

2x+2h -2x / h

2h / h

2

B)

1 / (x+h+1) - 1/(x+1) / h

[(x+1)/(x+h+1)(x+1) - (x+h+1)/(x+1)(x+h+1) ] / h

[ (x+1) / (x+h+1)(x+1) - (x+h+1) / (x+1)(x+h+1) ] / h

[-h / (x+1)(x+h+1)] / h

-1 / (x+1)(x+h+1)

C)

(x+h)^2 - (x^2) / h

x^2 + 2xh + h^2 - x^2 / h

2xh + h^2 / h

2x + h