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Understanding continuity in terms of limits

Page history last edited by PBworks 15 years ago

Understanding Continuity in Terms of Limits

 

Shelby Smith

 

 

A function is continuous if it is unbroken fro all values of x. The formal definition states that f(x) is continuous if the limit as f(x) approaches a is equal to that of f(a).

 

It is also important to remember that the lim of f(a) exists iff f(x) approaching a from the left is equal to the lim f(x) as x approaches a from the right.

 

 

 

This can also be defined:

 

 

 

Therefore there are three possible problems which cause discontinuity

 

I. f(a) is not defined

II. the limit may not exist

 

 

III. the limit may exist but is not equal to f(a)

 

 

Sample Questions

 

1.

 

 

Image edited from http://www.richland.edu/james/lecture/m116/functions/piecewise1.gif

 

 

What is the

 

What is the

 

 

What is the

 

 

Is the function continuous at f(-1)?

 

2. Suppose that f(x) is continuous at x=3 and that , which statements about f(x) must be true?

 

I. f(3)=17

 

II. 3 is in the domain of f(x)

 

III.

 

 

 

 

 

 3. Is      continuous at x=2?

 

 

 

 

 

 

 

4. Suppose that the function f(x) is continuous at x=2 and that f(x) is defined

 

 

 

What is the value of a?

 

a)2

b)3/4

c)1/2

d)5/8

e)2/3

 

 5. For what values of x is  discontinuous?

 

a)  x=0

b)  x=1

c)  x=-1

d)  x=1 and x=-1

e)  all real numbers

 

 

6.   is continuous for all real numbers EXCEPT:

 

a)  x=0

b)  x=1

c)  x=1 and x=-1

d)  x=-1

e)  x=2

 

 

Answers

1.) 2, 2, does not exist, no

2.) All three

3.) No, althought the limit exists, the function is not continuous

4.) c

5.) b

6.)a

 

 

 

 

For more info:

 

http://www.sparknotes.com/math/precalc/continuityandlimits/section3.rhtml 

 

Sources:

Questions 3 and 4 from calculus text book

Questions 5 and 6 from Amsco's AP Calculus AB/BC by Maxine Lifshitz. NY.2004.

Comments (5)

Anonymous said

at 1:41 pm on Feb 8, 2008

I thought your page was really good. The definitions limits as they relate to continuity were very straightforward and clear. I also thought the questions were good because they were simple, but still tested our knowledge of what was on the page. Good job.

Anonymous said

at 2:32 pm on Feb 8, 2008

Page looks good, everything seems very efficient with a good range of problems. You might want to proofread this a little (specifically the word fro in place of for) but all in all it looks good.

Anonymous said

at 8:00 am on Feb 21, 2008

Good explanations. I particularly like the discussion of what would make a function DIScontinuous. Good problems, and appropriately cited. You need to re-read the page for typos (there's one in the first or second line) and grammar. Good job!

nbarnett@students.maret.org said

at 2:38 pm on Feb 2, 2009

I liked your page! I thought the explanation was clear and easy to follow. The page wasn't too jumbled. I also thought your questions were very good. I had to think about them a little but they were still simple. I would suggest proof-reading the explanation, but other than that, I thought your page was great!

nbarnett@students.maret.org said

at 7:57 pm on Feb 2, 2009

ps. I also liked the variety in your questions. - Nora

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