Its em again, I really need help on this one?

Question

Ok it seems that im really stuck on this one question, I know how to these types of problems I just don't know what is the next step to complete the problem, can someone plz help me?

Given f(x)= square_root_of (x) , evaluate

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


This is as far as i get:

( square_root(x+h) - square_root(x) ) / h

I need to get rid of the "h" ... what do i do next???

Im really sorry Capitan Ahab , but im not sure im not following what you are trying to say.... =(

Once again the problem is;

Given f(x)= square_root_of (x) , evaluate

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

Answer 1

Doing this from first principals

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

Step 1: multiply the top and bottom by the conjugate of [sqrt(x+h)-sqrt(x)] which is [sqrt(x+h)+sqrt(x)]

= lim h ->0 [sqrt(x+h)-sqrt(x)] * [sqrt(x+h) +sqrt(x)] / [ h * ( sqrt(x+h)+sqrt(x))]

simplify the top

= lim h->0 [(x+h)-(x)] / [ h * ( sqrt(x+h)+sqrt(x))]

= lim h -> 0 h / [ h * ( sqrt(x+h)+sqrt(x))]

cancel h 's

= lim h -> 0 1/ ( sqrt(x+h)+sqrt(x))

You can notice that there is no problem substituting h = 0 here so do it

= 1/( sqrt(x)+sqrt(x))

= 1/ [ 2 * sqrt (x)]

Answer 2

Ahab is correct and the equation (f(x+h)-f(x))/h in the limit as h ->0 is the definition of a thing called the 'derivative (which, I suspect, you'll be studying soon )

The 'trick' here is to evaluate the function so that it looks like h(g(x) +w(x)h +......)/h(q(x))

Now divide out the h to get (g(x) + w(x)h + ......)/q(x) Now, as h ->0 the only term that remains is g(x) and your answer is g(x)/q(x)

Hope that helps.


Doug

Answer 3

This is the derivative of sqrt(x) by definition

sqrt(x) = x^1/2

so the answer is (1/2)*x^(-1/2) = (1/2)*(1/sqrt(x))

FOR SURE!


SAW YOUR FOLLOW - UP:

Read the guy two answers down for the first principles - sounds like you haven't studied derivatives yet, which means you'd have to solve it by brute force, as below!

(watch your parentheses)

Answer 4

captain ahab is right.
i dont remember the steps though.
the answer by definition is 1/(2. square root x)

Answer 5

Do it this way.

lim(h==>0) [f(x+h) - f(x)] / h, where f(x) = sqrt(x)

Multiply by the conjugate:

[sqrt(x+h) - sqrt(x)] [sqrt(x+h) + sqrt(x)] / {h [sqrt(x+h) + sqrt(x)]}

The top is like (a-b)(a+b) = a^2 - b^2:

[(x+h) - x] / {h [sqrt(x+h) + sqrt(x)]}

because [sqrt(x+h)]^2 = x+h

The x's on top cancel, leaving you with

h / {h [sqrt(x+h) + sqrt(x)]}

Now the h's cancel, leaving

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

Now let h go to zero, giving

1/ [2 sqrt(x)]

You can rewrite this as (1/2) x^(-1/2)

This is the derivative of f(x) = sqrt(x), which you derived using the "delta process".