how do i solve: f(x)=x find lim(h to 0) f(x+h)-f(x)/h?

Question

what are the steps to solve this?

Answer 1

Fix x and for h ≠ 0,

{f(x + h) - f(x)} / h

= {x + h - x} / h

= h / h = 1.

So the limit as h → 0 is 1.

Answer 2

This is an example of a problem that seems too easy!

Basically, just plug in all the values shown, do some algebra, and you should see that when h or h^k , k positive appears in a numerator, its lim as h->0 is 0.

for f(x) = x
f(x+h) = x+h
f(x+h)-f(x) = h
then
[f(x+h)-f(x)]/h = h/h = 1
so lim is 1

Seems too easy, doesn't it?

note that this is the constant slope of your line y=x.

Answer 3

This is the definition of the derivative. If you have already learned derivatives, you know the derivative of f(x) = x is f'(x) = 1. Otherwise,

lim h goes to 0 {(x+h) - x} / h = h/h = 1

Answer 4

lim [f(x + h) - f(x)] / h
h -> 0

Since f(x) is given to be x, then
f(x + h) = x + h, and our limit becomes

lim [x + h - x]/h
h -> 0

Note that x and (-x) cancel each other out. This leaves us with

lim [h]/h
h -> 0

h/h is equal to 1.

lim 1
h -> 0

And the limit of a constant (since the variable h has disappeared) is just the constant itself.

lim 1 = 1
h -> 0

So the answer is 1.

Answer 5

lim(h to 0) [f(x+h)-f(x)]/h
= lim(h to 0) [(x+h)-x]/h
= lim(h to 0) h/h
= lim(h to 0) 1
= 1.

Answer 6

take the derivative