quick question on limits, undefined or infinity as the answer

Bronn

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hi

im working through my maths lecture notes and i seem to be having a misunderstanding about when a limit is undefined or its infinite.

for example:


lim x->0 cosec (x)


lim x->0 1/sin (x)


= 1/0


I would have thought to label this limit as undefined as it shoots of to +- infinity at x=0, but the answer in the book is just '+infinity'. Is infinity and undefined interchangeable? i didn't think it was, or am i just missing something?
theres numerous times that I've run into this problem on a a few questions.

cheers
 
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Limits, limits, limits... There is a lot to understand.

1) There is no AT for a limit unless it is continuous.
2) I don't really know what an "undefined limit" is. The limit exists or it doesn't.
3) There is no such thing as a limit "at infinity". Infinity isn't a place. There is a finite limit or there isn't.
4) It is not appropriate to say, "The limit is infinite." One should say, "Increases without bound" or "There is no limit."

Okay, what do you think about your cosecant, now?
 
lim x->0 [1/sin (x)]

= 1/0

… the answer in the book is just '+infinity'.
If you correctly typed the limit statement above, then the book is wrong.

\(\displaystyle \displaystyle \lim_{x \to 0} csc(x) = \text{DNE}\)

DNE means Does Not Exist. This limit does not exist because csc(x) is not approaching a single fixed value, as x approaches zero from both the right and left. This is a double-sided limit statement.

\(\displaystyle \displaystyle \lim_{x \to 0^+} csc(x) = \infty\)

This is a one-sided limit statement, and the limit does not exist. It does not exist because csc(x) is not approaching a fixed value, as x approaches zero from the right. When you see a statement that says some limit equals infinity, that's an abbreviation for: "the limit does not exist because the function increases without bound". Your book ought to have explained this notation. No limit ever equals infinity because that would contradict the very definition of a limit (i.e., that the function approaches some fixed, limiting value).

\(\displaystyle \displaystyle \lim_{x \to 0^-} csc(x) = -\infty\)

This is a one-sided limit statement, and the limit does not exist. It does not exist because csc(x) is not approaching a fixed value, as x approaches zero from the left. When you see a statement that says some limit equals negative infinity, that's an abbreviation for: "the limit does not exist because the function decreases without bound".

If a limit exists, then the function is approaching a fixed value. If it's a double-sided limit, then the function must approach the same fixed value, as x approaches from both directions.

PS: Don't write 1/0 to represent infinity; simply write "infinity" or . The expression 1/0 is meaningless because division by zero is not defined.

PPS: I hope your book is not one of those "for-dummies" publications. Both of the books that I've seen in those series were horrible. :cool:
 
thanks. i understand all the points you guys made, i just wrote it out clunky.

yeah it seems the book is wrong then, because it is simply written as lim x-> 0, not lim x ->0+.


so is DNE and undefined the same thing in this case?



and nope, its my unis course notes ..... ( a maths bridging class)
 
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… is DNE and undefined the same thing in this case?
I've never seen the word 'undefined' used to denote that a limit does not exist, but, if 'undefined' is what your book writes, then yes, it means that the limit does not exist.
 
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