Intro to Limit: Infinity & Zero Cases

[vissh]

Introductory "Limit"

hello :D Started to study calculus and was on "limits". I got a little doubt :D
When the value of any limit approaches to Infinity.i.e when the numerator have a non zero number but the denominator gets a zero on putting the variable's value , is it called that the limit doesn't exist??
For eg , can u tell me what is value of following :-
1) lim_x->2^- x/(x-2)
2) lim_x->2⁺ x/(x-2)
3) lim_x->0 3/x
4) lim_x->5⁺ 6/(x-5)
5) lim_x->5^- 6/(x-5)

Know that's very basic But if u know pls guide ^.^
Thanks in advance (^_^)

 

[micromass]

It would help a lot if you could tell us what you think the answers are. And it would help even more if you could tell us why you think they are the answers...

 

[vissh]

hmm.ok this is what i think:- [I using "a" to represent infinity]
1)As x->2 from left side , (x-2) will be a very small -ve number and thus, its reciprocal will get a very big absolute value i.e. its -a[-infinity]and thus -a*(no.close to 2) = -a.
so,for this left handed limit my answer is -a[-infinity] and left handed limit exists.

2)As x->2 from right side , (x-2) will be a very small +ve number and thus, its reciprocal will get a very big absolute value i.e. its +a[+infinity]and thus +a*(no.close to 2) = +a.
so,for this left handed limit my answer is +a[+infinity] and right handed limit exists.

As 1 and 2 have different answers ,thus, lim_x->2 x/(x-2) doesn't exist.

3)Using same argument as above , the left handed limit will give -a and the right handed will give +a And thus , the right and left handed limit exists. But the books says these doesn't exist as the denominator approaches 0 while numerator doesn't .

So, can u clear it out for me now as i think u will get where i am getting wrong :)

 

[micromass]

All these limits are correct!
You do need to distinguish between "the limit exists" and "the limit is infinity". If the limit is + or - infinity, then the limit doesn't exist (even if the left and right hand limits are equal).
The only way for a limit to exist, is if left and right hand limits are equal and they are a real number. So, technically, an infinite limit doesn't exist. That's what the book is trying to say...

 

[vissh]

okz Understood :D Thanks a lot Micro (^.^) One last thing to ask .
Suppose we got a limit lim_x->a f(x)
And f(x) is a function which have a term in its denominator which gets 0 when we put the value of a and this term can't be removed , do we find the limit by using the method i used ?
Thanks again :)

Edit :- okzzz ^.^

 

[micromass]

Yes, your method is always the right one. Except when you're in a "0/0"-situation (i.e. if both the numerator and denumerator is 0).