Concept of a limit being infinity

[dustbin]

This is more of a conceptual question dealing with a homework problem than with the problem itself...

So I am asked to find various limits for the function (x²-9x+8)/(x³-6x²). All well, no problems... I am asked to find the left and right hand limits as x→0, which is -∞ for each. At the end of the problem I am asked to either state the limit as x→0 or state that the limit does not exist. Here is where I have a question.

When a one-sided limit is either positive or negative infinity, the limit does not exist. When the equations for the one-sided limits are written, they are not stating that the limits exists but rather using it as a convenient description of the function's behavior near that value of x and as an explanation for why the one-sided limit does not exist. *I base this off of what it says in my text* However, if the one-sided limits at that point both go to either positive or negative inifinity, then the normal limit exists and L= +/-∞.

I'm having trouble grasping why the one sided limits do not exist when the function values approach infinity but the normal limit does exist when each of the one sided limits approach infinity. I can understand why we can say that the limit is infinity as a description of the function's behavior even though this limit does not actually exist. However, why is it wrong to say that this limit does not exist?

 

[Dick]

If a real function has a limit, then the limit of the function of the function should be a real number. As you said, saying a limit is +/- infinity is just a description of the behavior of the function near the limiting point. It's not wrong to say the limit doesn't exist, but your grader may feel otherwise. That's not wrong either. They are just asking for the limiting behavior if there's not a true limit value.

 

[algebrat]

This is more of a conceptual question dealing with a homework problem than with the problem itself...

So I am asked to find various limits for the function (x²-9x+8)/(x³-6x²). All well, no problems... I am asked to find the left and right hand limits as x→0, which is -∞ for each. At the end of the problem I am asked to either state the limit as x→0 or state that the limit does not exist. Here is where I have a question.

When a one-sided limit is either positive or negative infinity, the limit does not exist. When the equations for the one-sided limits are written, they are not stating that the limits exists but rather using it as a convenient description of the function's behavior near that value of x and as an explanation for why the one-sided limit does not exist. *I base this off of what it says in my text* However, if the one-sided limits at that point both go to either positive or negative inifinity, then the normal limit exists and L= +/-∞.

I'm having trouble grasping why the one sided limits do not exist when the function values approach infinity but the normal limit does exist when each of the one sided limits approach infinity. I can understand why we can say that the limit is infinity as a description of the function's behavior even though this limit does not actually exist. However, why is it wrong to say that this limit does not exist?

It is not "wrong" to say if limit goes to infinity the limit does not exist. It depends on where we are at in how we consider things. I could imagine a text might start out in one section by saying limit exists only if it is finite. Then, in later chapters, they may soften the definition by wanting to know if the function goes to infinity.

Notice there are many ways for a function to not converge to a finite value. It could oscillate in a bounded way away from any finite value, like sin(1/x) (x->0). It could go to +/- infinity (either side), and, just to show you that being unbounded is not the same as going to infinity, consider |sin(1/x)/x|. It is unbounded but does not converge to infinity.

 

[dustbin]

Great. Thank you very much, algebrat and Dick. I was just massively confused since I was told that my thinking was flawed but was not given nor could I find an explanation pertaining to this type of limit. I can see why the correct answer in my course is that the limit is infinity, but I was confused as to why I was being told that it is incorrect to say that this limit does not exist. Your responses have cleared my confusion!

I thought about it for a few days, reread the text (Stewart) and searched in later chapters, read Apostol's material on limits, and searched on the web and couldn't find an answer. Thanks a ton for clarifying!