Calculus - limits

Date: 12/10/95 at 22:47:46
From: Anonymous
Subject: Calculus - limits

Question:
I need some help with this. We have gone far in this semester in calculus,
but I have forgot how to do this. Thankyou in advance for any suggestions
you may give.

Find all the discontinuities of the following function and label those 
that are removable.

Given this piece-wise function.

      |   _________ 
      | \/ x^4 + 1        if  x < 0
      |  
      |      7            if  x = 0
      |    
      |   _____5_____  
      |     _______       if  0 < x < 11
      |   \/ 25 + x 
      |  
f(x)= |
      |   __5__           if  x = 11
      |     6             
      |  
      |  __(x-6)(x-13)__  if  x > 11
      |      6x - 78
      | 


I know that you have to find certain conditions to make the function
continuous. There are several rules that must happen if a function 
is continuous.

1.  Limit must exist.
2.  Left limit must = right limit.
3.  f(c) = limit   (I think)

How do you find the discontinuities and if they are removable ?
      
Thanks again for any help.

Simon B. Smith

Date: 7/23/96 at 20:45:41
From: Doctor Jerry
Subject: Re: Calculus - limits

Your comments 1, 2, 3 at the end of your note are correct. 

At x=0, the function is 7 and both one-sided limits are equal to 1.  
If we are allowed to redefine the function to be 1 at 0, then we can 
make the slightly different function continuous at 0.  Otherwise, the 
function as originally defined at 0 is not continuous. 

At 11 both one-sided limits are equal to 5/6, which is the function 
value there.  Hence it is continuous.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/