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# Infinite limits intro

Here we consider the limit of the function f(x)=1/x as x approaches 0, and as x approaches infinity. Created by Sal Khan.

## Want to join the conversation?

• is there a way f(x) = 0, if infinity isn't a number? •   There is no finite number that you can plug into 1/x to give zero. If you think about it as an algebraic expression ( y = 1/x ), plug 0 in for y and try to solve for x.
0 = 1/x (multiply both sides by x)
0*x = 1
We know that this cannot happen, as 0 times anything is 0. Therefore this is equation has no solution.
• Is there a difference between a limit that does not exist and a limit that is undefined? If so can someone elaborate :) •  Good question! A limit that does not exist and a limit that is undefined mean exactly the same thing. Just like the slope of a vertical line: It does not exist, and it is undefined.
• One question that is bugging me:
let be c the value that x approaches to, where c is within [-infinity, +infinity] and x is a Real number. According how Real numbers are defined, there is no real number x >= +infinity. After Khans explanation, in order a limit is defined, the following predicate must be true: if and only if lim x->c f(x), then lim x->c+ f(x) = lim x->c- f(x). But since there is no x where x >= +infinity, a limit where x approaches to infinity is undefined.
In other words: There is no real number x, that can approach to infinity from both sides, therefore a limit where x approaches to infinity is undefined.
I hope someone can enlighten me. • Great question!!
A HUMAN once said:
“...it's very much like your trying to reach infinity. You know that it's there, you just don't know where-but just because you can never reach it doesn't mean that it's not worth looking for.”
― Norton Juster, The Phantom Tollbooth

So the point is beautiful you cannot show infinity on a graph, but you can at least talk about it and show it by some sort of an identifier, an image, or a symbol.

Its almost like: nobody can tell you what apple tastes like, you have to be given the apple to find out but at least the people who have eaten an apple in their lives can talk among themselves and still know what they are talking about!
- Vedanta, Indian religious(véros science) text

I hope that was of help!!
(1 vote)
• How can their be a infinity on both sides of the number line? If you have a large number on the positive side, their is two times that number from the amount of numbers from the positive value and the negative value, right? If so, does this mean that their are different sizes of infinity? How is that possible, if it is true, when infinity goes on forever? • Infinity can be mathematically defined as an unbounded quantity greater than every real number. (http://www.wolframalpha.com/input/?i=infinity) To picture that on a number line, it is the quantity or number that is greater than anything we can imagine. The number line just keep going and going, greater and greater. Now, if that is in the positive direction on the number line, what about in the negative direction? I can imagine a very negative number (or a negative number with a great absolute value) such as -1,000,000,000 but there is always a quantity that will be more negative, hence "negative infinity."

I would not say infinity has a "size," because once you quantify how big it is, there can be something bigger. Think of positive and negative infinity as describing the direction of the unbounded value. Say we have a light source at 0 on the number line. One photon travels in the positive direction. Another photon travels in the negative direction. How far will the photons travel? If we leave the time frame undefined, the photons will travel an infinite distance, but in opposite directions.

• This is a super long question, so please bear with me:
At , Sal guides us to "think about a limit as x approaches either positive or negative infinity." Similarly in my Calculus classes, I have been taught that when both x and y are approaching infinity it the limit can only reach positive OR negative infinity but not both, as is the case with y=x.
My question is, why can't we have a limit that goes from +x and -x to infinity and negative infinity, and likewise for the y values?
The only way that I could think of requires 3 assumptions.
1). We have a sphere with an infinitely large radius,
2). We can take the circumference of the sphere, and
3). If we were on the plane of the line of the circumference, it would be completely straight to us.

Okay, so we have the circumference of an infinite sphere, and it seems straight to us. This means that we can either go left or right, in the negative and positive directions respectively. Because of the infinite proportions of the line, as we move to the right x approaches infinity and as we go left x approaches negative infinity. Because the line is circular and connected, if we continue along the right side of the line we will eventually reach the spot which we called negative infinity, and vice versa for the negative as well.
Because x approaches infinity from the left and from the right, the limit exists: x-> ±infinity f(x) = infinity.

All that to say, one can take a limit that reaches infinity from both negative and positive directions with correct stipulations.

My question: Why can't we have a limit that goes from +x and -x to infinity, and likewise for the y values? Why can't we? I've always been taught that this is impossible, but I feel that this method is feasible. I ask you, the Khan Academy community, does this work, and what can we do with it? • How can we differentiate between limits for functions that are infinite vs those that DNE? • Actually, if you take 1/|x-2|, the limit is infinity, therefore the limit does NOT exist. Think of lim = infinity as a special case of the limit not existing. Consider this intentionally absurd statement (from W. Michael Kelley's Humongous Book of Calculus Problems): "the limit is that it's infinitely unlimited". Yeah, makes no sense. If the limit is infinity, it means there is no limit, because the value just keeps increasing without limit.
• If f(x) = 1/x, isn't it undefined when it approaches 0, how does it have a limit of infinity? • I have a doubt that whether left limit of function ([x^2]-1)/x^2-1 as x tends to 1 is infinite or limit does not exist or both the same?( where [.] represent greatest integer function.)   