Question

The limit
lim (√(25+h)-5)/h
h->0

represents the derivative of some function f(x) at some number a. Find f and a.

Answer 1

This is the derivative. Solve it and integrate result.

Answer 2

integrate the result*

Answer 3

I should clarify. You'll have to just observe what the function is based on the definition of the derivative. Namely,
\[f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}.\]

Answer 4

or if you know the definition of the derivative:
f`(a)=lim (f(a+h)-f(a))/h
here f(a+h)=sqrt(25+h)
f(a)=5

Answer 5

@Andras, you have to uniquely find \(f(x)\) and the value it is being evaluated, \(a.\) Also, please do not post full answers. It's outlined against in the Code of Conduct.

Answer 6

a=0

Answer 7

but f is my problem

Answer 8

Why do you think it's \(a=0?\)

Answer 9

i evaluated the limit

Answer 10

Sorry havent used the site for ages, but I dont think I have given full answer.

Answer 11

Maybe it would help to put this in better terms:
\[f'(a)=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}.\]
Can you figure out what \(a\) is now?

@Andras, yes, but I thought that was your intention previously. Sorry if I was wrong in presuming such.

Answer 12

a=5?

Answer 13

That's close. Try again.

Answer 14

√25

Answer 15

Nope. \(a\) is a positive number. Try this, equate the two:
\[\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}=\lim_{h \to 0}\frac{\sqrt{25+h}-5}{h}.\]

Can you see it now?

Answer 16

yes

Answer 17

So, what must \(a\) be?

Answer 18

square root of 25+h

Answer 19

Not quite. That's \(f(a+h).\) If \(f(a+h)=\sqrt{25+h}\), what is \(a?\)

Answer 20

square root of 25

Answer 21

Why do you think it's \(\sqrt{25}?\)

Answer 22

because i plugged in \[\sqrt{25+h}\] in f(a+h) ?

Answer 23

You can't plug in anything into \(f(a+h);\) it is already defined for you in \(f(a+h)=\sqrt{25+h}\). You have to find the \(a\) such that it's a true statement.

Answer 24

how can i find it?

Answer 25

Well, \(f(a+h)=\sqrt{a+h}=\sqrt{25+h}.\) Do you see what \(a\) must be in order for the statement to be true?

Answer 26

25?

Answer 27

Correct.

Answer 28

Now, what is \(f(x)?\)

Answer 29

can you give me a hint ?

Answer 30

Well, we've seen that \(f(a+h)=\sqrt{a+h}.\) So, what would that look like if you let \(x=a+h?\)

Answer 31

\[\sqrt{25+h}\]

Answer 32

Not quite. We're letting \(x=a+h\). So what would \(f(x)\) look like, since we know \(f(a+h)=\sqrt{a+h}?\)

Answer 33

\[\sqrt{25+x}\]

Answer 34

Nope. You're getting close. Imagine we put \(x\) where \(a+h\) is. Do you know what \(f(x)\) is now?

Answer 35

f (\[\sqrt{25+h}\])

Answer 36

Well, that's not quite it. We're trying to look for the function which looks like \(f(a+h)=\sqrt{a+h}.\) What does \(f(x)\) have to be?

Answer 37

can you give me examples ?

Answer 38

Yes. If we have \(f(a+h)=a+h\), then \(f(x)=x.\) Or, if we have \(f(a+h)=(a+h)^2\), then \(f(x)=x^2.\) Another example: If \(f(a+h)=e^{a+h},\) then \(f(x)=e^x.\)
Also, if \(f(a+h)=\sqrt[3]{a+h},\) then \(f(x)=\sqrt[3]{x}.\)

Answer 39

√x?

Answer 40

Correct! Great job.

Answer 41

so f(x)=√x, and a=25?

Answer 42

Yup.

Answer 43

thanks very very thanks

Answer 44

You're welcome! :D