# Need to get rid of the 'h' on the denominator

1. Apr 5, 2005

### UrbanXrisis

$$lim_{h->0}\frac{ln(2+h)-ln2}{h}$$

so I need to get rid of the 'h' on the denominator, but how can I do that?

2. Apr 5, 2005

### Data

$$\frac{\ln (2+h) - \ln(2)}{h} = \ln \left[\left( 1 + \frac{h}{2}\right)^{\frac{1}{h}}\right]$$

Do you see why?

Now, this may or may not help you, depending on how much you know. If you know

$$\lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n = e^x,$$

and

$$\lim_{x \rightarrow a} \ln f(x) = \ln \left( \lim_{x \rightarrow a} f(x) \right)$$

then this is enough for you to do the question.

Last edited: Apr 5, 2005
3. Apr 5, 2005

### UrbanXrisis

I really do not understand..

my book explains it as... $$lim_{h->0}\frac{ln(2+h)-ln2}{h}=f'(2)$$, where $$f(x)=lnx$$.

what does that mean?

4. Apr 5, 2005

### dextercioby

It means that the book applied the definition of the derivative of a function of one variable (which is $f(x)=\ln x$) at a point from its domain (which is $x=2$)...

It can't get any more clear/elementary than that.

Daniel.

5. Apr 5, 2005

### Data

It means that it is the derivative of the function $\ln{x}$ evaluated at $x=2$. A derivative can be thought of geometrically as a rate of change of one quantity with respect to another. In this case, it is the rate of change of the function $\ln x$ with respect to $x$, at the point $x=2$.

The derivative of a function $f(x)$ at the point $x$ with respect to $x$ is defined to be

$$f^\prime (x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$

if this limit exists.

The numerator of the limitand, $f(x+h)-f(x)$, is the change in $f$ between $x$ and $x+h$. The denominator is the change in $x$, which is just $h$. As the change in $x$ approaches zero (ie. when we take the limit), we get the instantaneous rate of change of the function with respect to $x$.

6. Apr 5, 2005

### what

Think of it as your good old rise over run to find the slope of a function. Slope is your rate of change, and Fermat(i think) came up with this neat way of computing the slope for a function that's not a line. You pick the point you want which is x then the difference between the point you want and any other point is h, so the denominator on the bottom is your run or difference in x, hence h on the bottom. Then you draw a secant line that crosses that function at the point x and another arbitrary point. The slope would then be (y-y1)/h, y1 is also f(x) in other notation and y is the function evaluated at your point x + your run h, or ((x+h),y), and in other notation f(x+h). Put it all together and you get your function, take the limit of that function as the run aproachers 0 and you get the derivative or slope at that point.

7. Apr 5, 2005

### dextercioby

Folklore says that Leibniz discovered diff.calculus while playing with tangents to curves...

Daniel.

8. Apr 5, 2005

### UrbanXrisis

okay,I understand all that you have said. f(x) is ln(x).

how do I actually solve the problem?

9. Apr 5, 2005

### MathStudent

just to play devils advocate ...

Sure, after he stole the idea from Newton! :tongue2:

10. Apr 5, 2005

### dextercioby

He didn't steal anything.Not even phylosophy and physics...He was a really brilliant man.In the same league with Newton,Euler & Gauss.

Daniel.

11. Apr 5, 2005

### SpaceTiger

Staff Emeritus
See how Data rearranged the limit to a different form in his first response? First, verify that you understand how he got it to that form. Then, see if you can use his second and third formulae to get a result.

12. Apr 5, 2005

### dextercioby

What do you mean...?It's already solved with the derivative's definition.

Daniel.

13. Apr 5, 2005

### MathStudent

I'm purely joking, as far as I know, Leibniz and Newton discovered the idea independently of each other .

14. Apr 5, 2005

### dextercioby

Not to mention in different context.Was was interested in maths,the other in physics...

Daniel.

15. Apr 5, 2005

### SpaceTiger

Staff Emeritus
If this is like the structure of my old calc class, he may not have covered derivatives yet.

16. Apr 5, 2005

### UrbanXrisis

I dont see have Data's second and third equation fit into all of this

17. Apr 5, 2005

### UrbanXrisis

I've learned derivatives, I think I just forget the definition of a derivative and how to solve the definition, it was a while back and my class is doing some basic review

18. Apr 5, 2005

### dextercioby

If u did,then u'd understand that

$$\lim_{h\rightarrow 0} \frac{\ln(2+h)-\ln 2}{h}=:\left \left(\frac{d}{dx}\ln x\right) \right|_{x=2} =\left\frac{1}{x}\right|_{x=2}=\frac{1}{2}$$

Daniel.

19. Apr 5, 2005

### SpaceTiger

Staff Emeritus
Well, if you're allowed to solve this problem by invoking the derivative of a logarithm, just do that. Otherwise:

$$\frac{\ln (2+h) - \ln(2)}{h} = \ln \left[\left( 1 +\frac{h}{2}\right)^{\frac{1}{h}}\right]$$

Explain to me how this is true so that I can be sure that you're following.

20. Apr 5, 2005

### UrbanXrisis

oh my, okay, thanks, I get it