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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 1

Lesson 19: Unbounded limits (vertical asymptotes)

# Analyzing unbounded limits: rational function

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.1 (EK)
,
LIM‑2.D.2 (EK)
Sal analyzes the behavior of f(x)=-1/(x-1)² around its asymptote at x=1.

## Want to join the conversation?

• can we say in that case the limit as x approach 1 will be negative infinity.. because the left and the right limit are the same ??
• We can (and often do) say that, but to be completely correct we should say something like "the limit as x approaches 1 diverges towards negative infinity".

The reason we aren't supposed to say a limit will be (negative) infinity, is that infinity is not a real number – it might be better thought of as a direction ...

If you are interested, there is a whole area of mathematics devoted to infinities!
• Throughout the video, is it incorrect for it to be saying that the limit of the function = positive or negative infinity? I ask because thinking the way it talks in this video made me think that both limits talked about in this video exist, which Sal says is wrong:
• Unbounded limits don't exist; however, they are different from limits such as a_n = (-1)^n ; this sequence doesn't have a limit merely because it is alternating between 1 & -1, though its absolute value stays at 1. Unbounded limits aren't oscillating - they keep getting bigger or smaller. So we define infinity & - infinity to represent that. Technically they aren't "real numbers" but they are apart of the extended real number system. A reason as to why the limits can't exist is because consider 1 = x*1/x (x > 0) as x approaches 0 from the right. If the limit existed we could write lim x * 1/x = lim x * lim 1/x = 0 * (infinity) = 0. But the limit is clearly 1. So saying the limit doesn't exist is just a reminder we can't use limit properties to pull apart operations.
(1 vote)
• how can i know the limit as x approaches 1 in the above function ?
(1 vote)
• - infinity.
That's exactly what was found out in the video, since both the one-sided limits = - infinity, the two-sided limit also is - infinity
(1 vote)
• is there any other way to do it except the estimating tabls way
• Yes, these techniques are covered in the rest of the limits playlist.
• Can I put a function and then you give me the answers of its limits step by step?
• Would you want me to find a limit here in the answer section? I would be glad to help.
• How can you put a sign on infinity? I can prove you can't given 1/∞=0.
x=x
x=x*x/x
x=(x^2)(1/x)
|x|=|x^2||1/x|
|x|=(x^2)|1/x|
|x|/x=|1/x|(x^2)/x
|x|/x=|1/x|x
|x|/x=|1/x|/(1/x)
Now substitute +-∞. You get:
∞/(+-∞)=|1/∞|/(1/+-∞)
We know that 1/∞=0, so 1/0=∞, so:
∞/(+-∞)=0/+-0
∞/(+-∞)=0/0
∞/(+-∞)=(1/∞)/(1/∞)
∞/(+-∞)=∞/∞
(+-∞)/∞=∞/∞
+-∞=∞
Positive and negative infinity are both actually just plain old infinity. Right?