Is the Limit Infinity or Does It Not Exist at a Vertical Asymptote?

In summary, when dealing with a function with a vertical asymptote that approaches infinity from both sides, the limit approaching from either side would be infinity. However, since infinity is not a real number, you can also say that the limit does not exist.
  • #1
Jimmy25
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0
This has been bugging for a while and I haven't found an answer.

Say you have a function with a vertical asymptote. This asymptote approaches infinity from both sides.

The limit approaching from either side would be infinity. So would you say the limit is infinity or does not exist?
 
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  • #2
Here's a function that does what you describe - f(x) = 1/x2.

[tex]\lim_{x \to 0} \frac{1}{x^2}~=~\infty[/tex]

In one sense, the limit does not exist, because infinity is not a number in the reals. What this limit is saying is that the closer x gets to 0 (from either side), the larger 1/x2 gets.
 
  • #3
Jimmy25 said:
This has been bugging for a while and I haven't found an answer.

Say you have a function with a vertical asymptote. This asymptote approaches infinity from both sides.

The limit approaching from either side would be infinity. So would you say the limit is infinity or does not exist?
You can say either one. "Infinity" is not a real number so saying that a limit is "infinity" (or "negative infinity) is just saying that the limit does not exist for a particular reason.
 

Related to Is the Limit Infinity or Does It Not Exist at a Vertical Asymptote?

1. What does it mean for a limit to approach infinity?

When we say that a limit approaches infinity, it means that the value of the function is getting larger and larger without bound as the input approaches a certain value. In other words, the function is growing without limit as the input gets closer and closer to a certain value.

2. How do you determine if a limit approaches infinity?

To determine if a limit approaches infinity, we need to evaluate the function at values that are increasingly closer to the given input value. If the function values keep getting larger and larger without bound, then the limit approaches infinity. However, if the function values approach a finite number, then the limit does not approach infinity.

3. Can a limit approach infinity from both sides?

No, a limit can only approach infinity from one side. This is because infinity is not a real number and therefore, it cannot be approached from both sides. We can only approach infinity from the positive or negative direction, depending on the given function and input value.

4. How is a limit approaching infinity different from a limit at infinity?

A limit approaching infinity refers to the behavior of a function as the input gets closer and closer to a certain value. On the other hand, a limit at infinity refers to the behavior of a function as the input approaches infinity, or in other words, when the input gets larger and larger without bound.

5. Can a limit approaching infinity have a finite value?

No, a limit approaching infinity cannot have a finite value. This is because infinity is not a real number and therefore, a function cannot have a finite value at infinity. If the limit of a function approaches infinity, it means that the function is growing without bound and does not have a finite value.

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