If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: AP®︎/College Calculus AB>Unit 1

Lesson 14: Connecting infinite limits and vertical asymptotes

# Introduction to infinite limits

Introducing the notation of infinite limits.

## Want to join the conversation?

• Just curious, but is there a such thing as the opposite of infinity? What I mean is, since infinity is the notion of an incomprehensibly large number that doesn't follow the rules of arithmetic, is there a such thing an an infinitely tiny, minute, incomprehensibly small number?
• Yes, there are systems with a reciprocal of infinity, such the hyperreal numbers and the surreal numbers.

The hyperreals are basically the reals with 'infinte' and 'infinitesimal' elements appended. The surreals have a more complicated construction, which is walked through in Donald Knuth's novel 'Surreal Numbers'.
• how do you know where the line is coming from and where it is going?
• In these scenarios, you can't just plug in the value because the values approach an asymptote. If the line is coming from the negative side/left and plunging down rather than showing a value, it is going infinitely down, or to negative infinity. Same thing goes for the other side.
• if i say that the limit is infinity and not unbounded , does it help me in math problems ?
• There are many forms of indeterminate forms and they include forms of infinity. Further we will encounter problems having lim of x tends to infinity where we will have to work ourselves out.
Thus, for the starters, most of the teachers prefer to avoid the concept of infinity, but the concept itself is very versatile in my opinion.
• One could still say a limit is unbounded or does not exist for a function that grows without bound in a positive direction on one side of an asymptote and a negative direction on the other side, correct?

So am I right in thinking this notation just gives us a more nuanced way to talk about unbounded one-sided limits and limits that grow without bound in the same direction on both sides of an asymptote?
• Can't we say $\lim_{x \to 0} x^{-1} = \pm \infty$? Is there some shorthand mathy notation for “limit does not exist/is unbounded”?
• First of all, there's unfortunately no LaTeX here on KA.
"Does not exist" is either ¬∃ (\neg\exists), ~∃, or exists with a slash through it (sorry I couldn't find the symbol, but it's \nexists). Unbounded would just be written out as infinity or the text "is unbounded".
However, in this case, you cannot say that the limit is unbounded. It simply does not exist. If the left hand limit does not equal the right hand limit, or the limit oscillates between two values, you can only say that it is nonexistent.
Let me know if this helps.
• Didn't we have a previous lesson where evaluating these exact situations as infinity wasn't allowed? Maybe I'm thinking of an IRL class I took, but I remember going on at length about why we wouldn't use infinity as the answer to such questions.
• Technically, yeah, saying the limit is infinity is not really correct. Infinity is really just a shorthand notation for a much more rigorous definition of the limit, which you'll learn later as the "epsilon-delta (ε-δ) definition" (or in this case, the M-δ definition. Why M? It's because ε and δ are incredibly small numbers, while M here is supposed to be considerably big. So, we changed notations). I'll provide the rigorous definition below.

If I have lim (x-->a) f(x) = infinity, this, more rigorously, means that for some M > 0, there exists a δ > 0 exists such that |f(x)| ≥ M implies |x-a| ≤ δ. If I need to tone down the math lingo here, it basically says that no matter how big you make your value of M, there will always be a value of x, which will be δ units away from a (either to the left or right of a. hence, the absolute value is used), which will give an f(x) value greater than M.

This probably won't make much sense now, but once you learn about epsilon-delta, it'll start to make more sense.
• What is the limit of x to the xth power as x approaches 0?
lim_(𝑥→0)⁡〖𝑥^𝑥 〗
• Technically the limit does not exist as the limit from the left side is not well-defined. However, the limit of x^x as x approaches 0 from the right is 1.
• When Sal mentions negative infinity vs. positive infinity, I understand we are talking about the direction of f(x) as x increases or decreases. However, there is no such thing as negative infinity. Infinity is infinity, so if it is negative or positive in direction why does that matter? Mathematically, it should be the same, right?
• No, it is not the same. Negative infinity is when a number gets infinitely negative (like -1, -2, -3, -4...) and positive infinity is when a number gets infinitely positive (1, 2, 3, 4...). As you can see, they are not the same. If a function approaches positive infinity, this means that it goes, colloquially speaking, "up". If the function approaches negative infinity, it goes "down". Hope this helps!