Simple limit problem (Which way is the best to solve?)

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In summary, it is not possible to solve this limit analytically because the denominator is not well-defined on the left side. The best approach would be to use a calculator to approximate the limit by plugging in values close to 3. If the numerator has a well-defined non-zero limit and the denominator does not, then the limit does not exist. It is not necessary to provide a proof in this case. It is important to keep in mind the behavior of the function and its domain when evaluating limits.
  • #1
lLovePhysics
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I'm wondering what the best way is to solve:

[tex]\lim_{x \rightarrow 3^{-}} \frac{x}{\sqrt{x^2-9}}[/tex]

I'm pretty sure that f(x) is not equal to zero but I can't seem to manipulate it to cancel out (x-3). Also, when solving these types of problems, can you use the same rules as regular limits and substiute accordingly to obtain the limit as x approaches 3 from the right or left? Do people usually solve these problems analytically or graphically? I guess if you understand the function and can visualize it and know the laws and whatnot you can solve it analyitically right?

In a nutshell, what is the simplest method into solving these problems accurately and quickly? (One that provides enough proof)
 
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  • #2
What I'm thinking now is that since it is the limit as x approaches 3 from the left, you can just solve for the limit. However, if the limit does not exist, then you should revert back to the continuity rules or graph the function to solve it?
 
  • #3
Ask yourself first what do the numerator and denominator approach separately? This may be less of a problem than you think.
 
  • #4
If the limit doesn't exist, as I think you know, then it doesn't exist. You can't 'solve it'.
 
  • #5
Well the numerator approaches 3 from the left and the denominator approaches 0 from the left so would it be -3? How are you suppose to support your answer though? I'm learning continuity right now so am I suppose to use those rules?
 
  • #6
Best way would be to try and "pull out" the x^2 inside of the square root and try to cancel that x on the numerator.
 
  • #7
bob1182006 said:
Best way would be to try and "pull out" the x^2 inside of the square root and try to cancel that x on the numerator.
my algebra isn't that good, how would you do that? i saw a crazy one today ... I'm still trying to figure it out :(
 
  • #8
lLovePhysics said:
Well the numerator approaches 3 from the left and the denominator approaches 0 from the left so would it be -3? How are you suppose to support your answer though? I'm learning continuity right now so am I suppose to use those rules?

Pull out your calculator and start picking numbers close to 3 to approximate the limit (aside from the fact that in the left limit the denominator is not really well defined) and tell me how we should assign the limit? If the numerator has a well defined non-zero limit and the denominator doesn't there is no limit. I really don't think this calls for a 'proof'. As I said the denominator isn't even well defined. Proof enough for me.
 
  • #9
lLovePhysics said:
Well the numerator approaches 3 from the left and the denominator approaches 0 from the left so would it be -3? How are you suppose to support your answer though? I'm learning continuity right now so am I suppose to use those rules?

Since when is 3/0 anything close to -3?
 
  • #10
Oh, ok so it approaches -infinity right? I'm assuming you can't solve this analytically? Yeah bob, how can you pull the x^2 out of the sqrt?
 
  • #11
rocophysics said:
my algebra isn't that good, how would you do that? i saw a crazy one today ... I'm still trying to figure it out :(

[tex](ax^b + c)^\frac{d}{e} = x^\frac{bd}{e}(a+\frac{c}{x^b})^\frac{d}{e}[/tex]

@ILovePhysics hm..Is the denominator even defined on the left side? I graph it and it exists but can't find a formula equivilant o_O

and yep you're correct -infinity is what it goes too, somehow o.o.
 
  • #12
Dick said:
Since when is 3/0 anything close to -3?

Well, I just thought there was an asymptote at -3 but my brain wasn't functioning well and I just picked -1 for the denominator lol.
 
  • #13
lLovePhysics said:
Oh, ok so it approaches -infinity right? I'm assuming you can't solve this analytically? Yeah bob, how can you pull the x^2 out of the sqrt?

1/x can be said to approach -infinity as x->0^- in some sort of reasonable sense. It's also reasonable to say the limit doesn't exist. This is worse. What is sqrt(-0.0001)?
 
  • #14
bob1182006 said:
[tex](ax^b + c)^\frac{d}{e} = x^\frac{bd}{e}(a+\frac{c}{x^b})^\frac{d}{e}[/tex]

@ILovePhysics hm..Is the denominator even defined on the left side? I graph it and it exists but can't find a formula equivilant o_O

and yep you're correct -infinity is what it goes too, somehow o.o.

It is not defined, on that you are correct. Tough to get a limit then.
 
  • #15
Oh yeah, I forgot that infinities are never limits; is it because they are "infinity" and have no limit? It seems intuitive to me now. Thanks for your help guys!
 
  • #16
lLovePhysics said:
Oh yeah, I forgot that infinities are never limits; is it because they are "infinity" and have no limit? It seems intuitive to me now. Thanks for your help guys!

Right. It's because the answer to a limit question is generally supposed to be a real number. Infinities aren't real numbers. They definitely don't work in an epsilon-delta sense.
 
  • #17
Dick said:
It is not defined, on that you are correct. Tough to get a limit then.

The interval -3 < x < 3 isn't in the domain of the function, so this is worse than just "limit does not exist" -- the expression is just meaningless. I suspect this was a trick question given to the students to see if they are watching out for the behavior of the function.
 
  • #18
It's because "infinity" is not a real (or complex) number. Limits are always numbers. Saying "the limit is infinity" is just saying the limit does not exist for a specific reason.
 
  • #19
HallsofIvy said:
It's because "infinity" is not a real (or complex) number. Limits are always numbers. Saying "the limit is infinity" is just saying the limit does not exist for a specific reason.

Those last two statements aren't quite right.

An example of a limit that is *not* infinity, but rather does not exist is lim x-> infinity of sin x. Sine is a bounded function, but sin x does not approach any particular value between -1 and 1 more closely as x grows without limit.

Another example is lim x-> 0 of ln x. The limit as x approaches 0 "from above" is minus infinity, but values x <= 0 are not in the domain of ln x, so the limit from below does not exist because the function has no definition there. So the "two-sided" limit does not exist because there is no way to define the limit (or the function) for x<0.

Even for a function where the limit at a "from above" differs from the limit at a "from below" , the "two-sided" limit at a would be said not to exist. The individual "one-sided" limits can still each be finite and this would still be the case.

What you are describing is referred to as an "infinite limit". An example would be
lim x-> 0 of (1/(x^2)). Each of the "one-sided" limits at x = 0 is positive infinity, so the "two-sided" limit is said to be positive infinity. The limit *does* exist, but it is not a number. [The same cannot be said for lim x-> 0 of (1/x), because the "one-sided" limits are *differently-signed* infinities. So the "two-sided" limit at x=0 in this case does not exist.]

"Infinity" is not used simply as a place-holder to signify the failure of a function to have a limit. There is a formal definition for an infinite limit which does place certain requirements on the behavior of the function.
 
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  • #20
Thanks. Most illuminating.
 
  • #21
Dick said:
Thanks. Most illuminating.

I'm glad if you find it helpful. (I am presently teaching this to Calculus I students, so definitions of limits are much on my mind at the moment...)

The issue with this particular problem is that they wanted the limit as x-> 3 "from below". If we are restricted to real-valued functions, then the denominator does not even have a defined value, so you're done right there.
 
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  • #22
HallsofIvy said:
It's because "infinity" is not a real (or complex) number. Limits are always numbers. Saying "the limit is infinity" is just saying the limit does not exist for a specific reason.

dynamicsolo said:
Those last two statements aren't quite right.

An example of a limit that is *not* infinity, but rather does not exist is lim x-> infinity of sin x. Sine is a bounded function, but sin x does not approach any particular value between -1 and 1 more closely as x grows without limit.

Another example is lim x-> 0 of ln x. The limit as x approaches 0 "from above" is minus infinity, but values x <= 0 are not in the domain of ln x, so the limit from below does not exist because the function has no definition there. So the "two-sided" limit does not exist because there is no way to define the limit (or the function) for x<0.

Even for a function where the limit at a "from above" differs from the limit at a "from below" , the "two-sided" limit at a would be said not to exist. The individual "one-sided" limits can still each be finite and this would still be the case.

What you are describing is referred to as an "infinite limit". An example would be
lim x-> 0 of (1/(x^2)). Each of the "one-sided" limits at x = 0 is positive infinity, so the "two-sided" limit is said to be positive infinity. The limit *does* exist, but it is not a number. [The same cannot be said for lim x-> 0 of (1/x), because the "one-sided" limits are *differently-signed* infinities. So the "two-sided" limit at x=0 in this case does not exist.]

"Infinity" is not used simply as a place-holder to signify the failure of a function to have a limit. There is a formal definition for an infinite limit which does place certain requirements on the behavior of the function.

I agree with everything but your last two paragraphs! Since a limit is by definition a number, it makes no sense to say "The limit *does* exist, but it is not a number." Every text I have seen agrees that if "the limit is infinity" then the limit does not exist.
 
  • #23
Heres an example of a formal, delta-M definition of limits involving infinity
http://img513.imageshack.us/img513/7637/limitts5.png [Broken]
 
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  • #24
HallsofIvy said:
Since a limit is by definition a number, it makes no sense to say "The limit *does* exist, but it is not a number." Every text I have seen agrees that if "the limit is infinity" then the limit does not exist.

I suspect there may have been some controversy at one time as to whether a limit was required to be a number in order to exist. However, as turdferguson points out, there is a formal approach for dealing with infinite limits which is an extension of the epsilon-delta definition for finite limits; and many texts I've seen do use it.
 

1. What is a simple limit problem?

A simple limit problem is a mathematical problem that involves finding the value of a function as the independent variable approaches a specific value, typically represented as x → a. The answer to a simple limit problem is the limit or the value that the function approaches at that particular value of x.

2. What are the different ways to solve a simple limit problem?

There are several methods to solve a simple limit problem, including direct substitution, factoring, rationalization, and the use of trigonometric identities. The method you choose depends on the type of function and the given limit.

3. How do I know which method to use for a specific simple limit problem?

The method you use to solve a simple limit problem depends on the type of function and the given limit. If the limit involves a polynomial function, direct substitution is usually the best method. If the limit involves a rational function, factoring or rationalization may be necessary. If the limit involves trigonometric functions, the use of trigonometric identities may be helpful.

4. What are some common mistakes to avoid when solving a simple limit problem?

One common mistake when solving a simple limit problem is to forget to check if the function is continuous at the given value of x. If the function is not continuous, the limit may not exist. Another mistake is to use the wrong method or make a calculation error. It is important to carefully consider the given function and limit and choose the appropriate method to solve the problem.

5. Are there any tips for solving simple limit problems more efficiently?

One tip for solving simple limit problems more efficiently is to simplify the function as much as possible before attempting to evaluate the limit. This can help to avoid complicated calculations and make the problem easier to solve. Another tip is to practice, as solving more problems will improve your understanding and speed when it comes to solving simple limit problems.

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