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September 24th, 2011, 01:53 PM   #1
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Limit statements

which of the following statements MUST be true, MIGHT be true, or is NEVER true about a function f which is defined for all real numbers? Justify your answer.
correct me if I am wrong.

a) lim x->a f(x)=f(a) might be true if this is a continuous function

b)lim x->0 f(x)/x=1 then f(0)=0 might because???

c)If lim x-> f(x)/x=1 then lim x->0 f(x)=0 must be true because the hole is at 0, so f(x)=0
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September 24th, 2011, 03:00 PM   #2
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Re: Limit statements

Quote:
Originally Posted by layd33foxx
which of the following statements MUST be true, MIGHT be true, or is NEVER true about a function f which is defined for all real numbers? Justify your answer.
correct me if I am wrong.

a) lim x->a f(x)=f(a) might be true if this is a continuous function

b)lim x->0 f(x)/x=1 then f(0)=0 might because???

c)If lim x-> f(x)/x=1 then lim x->0 f(x)=0 must be true because the hole is at 0, so f(x)=0
a) must be true (definition of continuous function)
b) if f(0) not 0 (assuming continuity), then the limit is infinite, so f(0)=0 is necessary, although not sufficient.
c) seems to have a typo: x -> 0? question seems to be a slight variation of b).
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September 24th, 2011, 04:03 PM   #3
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Re: Limit statements

c)If lim x->0 f(x)/x=1 then lim x->0 f(x)=0
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September 25th, 2011, 05:24 AM   #4
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Re: Limit statements

b) might
c) must

Isn't saying "... Is necessary" the same as saying "must"?
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September 25th, 2011, 02:04 PM   #5
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Re: Limit statements

For b) must. I don't see any real difference between b) and c).
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September 25th, 2011, 02:59 PM   #6
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Re: Limit statements

What about f(x) = 7 when x = 0,
= x otherwise? The limit as x - > a (for all real a) is 1...
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September 26th, 2011, 10:00 AM   #7
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Re: Limit statements

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Originally Posted by mathman
For b) must. I don't see any real difference between b) and c).
b) refers to the limit of the function f at x = 0
c) refers to the value of the function f at x = 0

That b) isn't "must" can also be seen by considering f(x) = e^x.
Then f(x)/x = e^x/x, which has limit of 1 as x approaches 0 (by L'Hospital). Yet f(0) = 1.
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September 26th, 2011, 12:50 PM   #8
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Re: Limit statements

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Originally Posted by The Chaz
Quote:
Originally Posted by mathman
For b) must. I don't see any real difference between b) and c).
b) refers to the limit of the function f at x = 0
c) refers to the value of the function f at x = 0

That b) isn't "must" can also be seen by considering f(x) = e^x.
Then f(x)/x = e^x/x, which has limit of 1 as x approaches 0 (by L'Hospital). Yet f(0) = 1.
e^x/x - L'Hopital's rule doesn't apply, since it is not indeterminate. It becomes 1/0 as x ->0
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September 26th, 2011, 01:14 PM   #9
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Re: Limit statements

Ah I got carried away!
What about x^2/x ?
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