Relating limits to derivatives, as x approaches non zero number?

[i_m_mimi]

Homework Statement

Suppose that f' (2) = 3. Find the limit as x approaches 2 of [f(x)−f(2)]/[sqrt(x) - sqrt(2)]

Answer: 6*sqrt 2

Homework Equations

The Attempt at a Solution

f'(x) = lim h->0 = [f(a +h) - f(a)]/h = slope [f(x)-f(2)]/ x-2

a = 2

i would think that the limit = 3? wrong answer

tried graphing it, but no original function is given
sqrtx and sqrt2 confuses me and not sure how to do it.

thank you

 

[SammyS]

Homework Statement

Suppose that f' (2) = 3. Find the limit as x approaches 2 of [f(x)−f(2)]/[sqrt(x) - sqrt(2)]

Answer: 6*sqrt 2

Homework Equations

The Attempt at a Solution

f'(x) = lim h->0 = [f(a +h) - f(a)]/h = slope [f(x)-f(2)]/ x-2

a = 2

i would think that the limit = 3? wrong answer

tried graphing it, but no original function is given
sqrtx and sqrt2 confuses me and not sure how to do it.

thank you

Rationalize the denominator of $\displaystyle \ \frac{f(x)−f(2)}{\sqrt{x\,}-\sqrt{2\,}}\ .$

 

[HallsofIvy]

Equivalently, write
$$\frac{f(x)- f(2)}{\sqrt{x}- \sqrt{2}}= \frac{f(x)- f(2)}{x- 2}\frac{x- 2}{\sqrt{x}- \sqrt{2}}$$
$$= \frac{f(x)- f(2)}{x- 2}\left(\frac{1}{\frac{\sqrt{x}- \sqrt{2}}{x- 2}}\right)<br /> <br /> \text{ and think about the derivatives of f(2) and }\sqrt{x}$$ at x= 2.

 

[i_m_mimi]

Equivalently, write
$$\frac{f(x)- f(2)}{\sqrt{x}- \sqrt{2}}= \frac{f(x)- f(2)}{x- 2}\frac{x- 2}{\sqrt{x}- \sqrt{2}}$$
$$= \frac{f(x)- f(2)}{x- 2}\left(\frac{1}{\frac{\sqrt{x}- \sqrt{2}}{x- 2}}\right)<br /> <br /> \text{ and think about the derivatives of f(2) and }\sqrt{x}$$ at x= 2.

I think I've got it :) how did you create those math symbols and format it that way? was it through the physicsforums website coding or by a program?


f'(2) = 3
x = 2

lim x->2 [f(x) - f(2)]/(x - 2) * lim x->2 1/ [sqrtx - sqrt2]/(x-2)

= (3) * 1/f'(sqrtx)
= 3 * 1/(1/2sqrtx)
= 3 * 2sqrtx
= 3 * 2 sqrt2
= 6 sqrt2

 

[Simon Bridge]

It's called $\LaTeX$ ;)

Underneath each post you will see a button marked "quote" - if you click on it, you will be taken to an advanced editor screen where everything in the post is included in "quote" tags. Read what is in the quote to see how all the symbols came out nicely formatted ;)

Use double-hash or double-dollar signs around the block of math you want formatted - and use a backslash "\" in front of the bits that need special formatting - so \sqrt{ \frac{\alpha}{\beta} } will come out $$\sqrt{\frac{\alpha}{\beta}}$$

 

[Mark44]

There's more information about using LaTeX here: https://www.physicsforums.com/showpost.php?p=3977517&postcount=3