Using the Definition of Derivative

Date: 9/1/95 at 21:31:15
From: Luke White
Subject: Calculus Homework- HELP!

To whom it may concern:

 I am a high school senior at St. Joseph's High School in
South Bend, Indiana, and I am currently enrolled in BC Calculus.
Below are a few problems that you may be able to help with. I would be 
extremely grateful. Thank you!

1.  Using the definition of derivative, find f'(x), in simplest form, 
for f(x)=x^(1/3). (I know through the "tricks" that the answer is 
(1/3)*x^(-2/3) but I can't get to that using the definition of 
derivative.)

2. If f(x)= [x], prove or disprove that f(x) has a limit at x=1

Thank you very much for your time and effort on this! I am exceedingly 
grateful.

With much thanks,
Luke White

Date: 9/4/95 at 11:38:32
From: Doctor Ken
Subject: Re: Calculus Homework- HELP!

Hello!

>1.  Using the definition of derivative, find f'(x), in simplest form, for 
>f(x)=x^(1/3).  (I know through the "tricks" that the answer is 
(1/3)*x^(-2/3) 
>but I can't get to that using the definition of derivative.)
>
Hint:  When you have this-

      x^(1/3) - (x+h)^(1/3)
Lim   ---------------------
h->0           h

try using the difference of two cubes, and multiplying the top and 
bottom by x^(2/3) + x^(1/3)(x+h)^(1/3) + (x+h)^(2/3).  It works in 
such a neat way!

>2. If f(x)= [x], prove or disprove that f(x) has a limit at x=1

Well, there are several ways that something can fail to have a limit 
as it approaches a point:

a) the point could be completely isolated, with no place to approach from.

b) the point could have approaches on both sides but they could be different.

c) the point might have really really messy approaches (ones that you 
couldn't draw with a normal pencil from Earth) and not even be 
continuous in a neighborhood of the point.

Basically, the point has to be on a graph where the left side and the 
right side both zoom in on the limit, and it's the SAME LIMIT ON BOTH 
SIDES.  Think about that in the context of this problem.  Is your 
teacher looking for a formal (delta - epsilon style) proof, or something 
a little less formal?

- Doctor Ken,  The Geometry Forum