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

Lesson 14: Connecting infinite limits and vertical asymptotes

# Analyzing unbounded limits: rational function

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!
• i cant believe the last comment on this video was 3 years ago.... well i better update that ig
• let me see...i'm going through and answering questions..nut sure this is a question...
• LETS SAY... x WAS! EQUAL TO 1 dun dun duoooooon.
Wa will happon
(1 vote)
• If x was equal to one the function will be undefined. The whole reason while we use limits is to approximate the values of functions at undefined points so making it equal to one negates that concept. This graph shows the graph of the function given in this video:https://www.desmos.com/calculator/ou6ywa7vsi
• 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:
(1 vote)
• 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.
• how can i know the limit as x approaches 1 in the above function ?
• - 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
• 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.
• Hi do you like to do math
• 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?