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## Precalculus

### Course: Precalculus>Unit 10

Lesson 14: Connecting limits at infinity and horizontal asymptotes

# Limits at infinity of quotients (Part 2)

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.3 (EK)
,
LIM‑2.D.4 (EK)
,
LIM‑2.D.5 (EK)
Sal analyzes the limits at infinity of three different rational functions. He finds there are three general cases of how the limits behave. Created by Sal Khan.

## Want to join the conversation?

• I understand how he gets to the limits and all, but can someone give me a 100% accurate definition of what exactly a "asymptote" is? what is an asymptote? Is it just the graph of the limit? please help me !
• This is actually not quite accurate.

Technically an asymptote to a curve is a line such that the distance between the curve and the line approaches zero as they both tend toward infinity.

A function can never cross a vertical asymptote, but it may cross a horizontal asymptote an infinite number of times (thus reaching it an infinite number of times). For example observe the limit of sin(x)/x as x approaches infinity.
• Are + and - infinity like approaching from the positive and negative sides? So limits approaching infinity dont exist, because + and - infinity yield different values?
• Not exactly,

positive and negative infinity represent the opposite "ends" of the number line. And here, "ends" is in quotation marks because the number line NEVER actually ends, it goes on forever in both directions. Basically positive infinity means to keep going towards bigger and bigger positive numbers. Think of the biggest positive number you can think of, and then go even bigger than that… and keep doing that… FOREVER! That's positive infinity.

For negative infinity, think of the most negative number you can think of, and then think of an even more negative number, and keep doing that, FOREVER.

So you see, if a limit approaches positive infinity from one side, and negative infinity from the other side… it doesn't approach the same thing from both sides. THIS is why the limit doesn't exist. It would be the same as saying that a limit that approaches 3 from the positive side and 2 from the negative side also doesn't exist. In order for a limit to exist it must approach the same thing from both sides.
• Where exactly can we use the concept of limits in our life?
In what kind of situations in our life do we need to find limits?
Please give me a real life example where in we have to find the limit.
• Limits are not that helpful in everyday life. However, as you study Calculus, you'll see that every single concept in Calculus in based upon limits. Derivatives are defined using limits, we can find the area under a curve, volume, surface area, arc length, radius of curvature using integrals that are all based upon limits. Without limits, there is no Calculus.
• In the solutions manual of my Calculus textbook, it gets the answer using a slightly different method. It divides like every term in the numerator and the denominator by the highest degree i guess and does all these weird calculations and then gets the answer. I understand this method much easier though (getting the highest power of both the numerator and denominator and then applying the x-> component) I guess my question is if I do this method that you are teaching on a test, is it still valid and legitimate? Of will i have to provide more work to justify my answer? Sorry for the incredibly long comment. Thanks.
• Sal's method is more of a "rationalize it" (no pun intended) approach. What your textbook says is the method I prefer, mainly because it gives you a clear, algebraic way to evaluate the limit.
As for which method to use on a test, that is something you should ask your teacher.
• So then would it be safe to assume that any time the numerator is growing faster than the denominator it will equal infinity, and any time the denominator is growing faster than the numerator it will equal 0? Or is there an exception?
• ``Notice that we are dealing with x nearly equal to Infinity. ``

Which is very very big. Thus we can safely assume.

But in case of small numbers like 2, 56, 345.... We cant assume so.
You can try it out yourself.
• At how can he simplify that to 1/2x when there are two different powers?
• He cancelled out common factors
3x^3/6x^4 can be factored to
(3*x*x*x)/(2*3*x*x*x*x)
You can cancel out the three and three of the x terms
( 3*x*x*x )/(2* 3*x*x*x *x)
This leaves 1/(2*x)
• at , is the simplification of (4/250)x to (2/125)x valid ?
• Yes, but it is unnecessary. With limits, since you often have them diverge toward +∞ or −∞ or else tend toward 0, you can save yourself unnecessary work by not simplifying any constants until you know you don't have an infinity or zero situation. When tending toward 0, your constant is irrelevant and there is no need to simplify. When tending toward ∞, you need only determine the sign of your constant, to determine whether you're tending toward + or − infinity.

It is only when you're tending toward a non-zero but finite number that you need to simplify your constant.

Note: The above applies to real numbers. If imaginary numbers get involved, the considerations can be more complicated.
• Lim(x--->infinity) (1+1/n)^n =e
Lim(x--->infinity) (1+1/x)^1/x=e
How to do solve these very strange infinity limits or either to prove them they equal to e?
• ``x		(1 + 1/x)^x	(1 + 1/x)^(1/x)1		2.000000000	2.0000000000000010		2.593742460	1.00957658300000100		2.704813829	1.000099508259151000		2.716923932	1.0000009995008310000		2.718145927	1.00000000999950100000		2.718268237	1.000000000100001000000		2.718280469	1.0000000000010010000000	2.718281694	1.00000000000001``
• Will a similar method be used for an expression in which there is no denominator like for limit x tending to infinity √(n-1) - √n
Here will it be correct to say that this is the same as √n - √n (As the 1 will be negligible when n tends to infinity) and thus the limit equals 0 ?
• Yep. That general pattern of thinking is correct; actually, though, the example you've mentioned is a bit interesting in the sense that we can't always define a limit that takes the form ∞ - ∞. In this case, because the two terms are of the same degree, the limit is equal to 0 (and a quick glance at the graph of `y = sqrt(x-1) - sqrt(x)` confirms that as x approaches infinity, y approaches 0). As you said, it resembles `y = sqrt(x) - sqrt(x) = 0` in the limit.

Other limits of a similar nature may not always behave the same way. Take the limit of `x^3 - x^2` as x approaches infinity, and we get infinity rather than 0 because the terms are of a different degree (which seems fairly clear just by looking at the function). Sometimes the examples are less clear-cut, so it's worth exercising some caution with limits of the form ∞ - ∞.

I hope you find that helpful; you're definitely on the right track.