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## AP®︎ Calculus BC (2017 edition)

### Course: AP®︎ Calculus BC (2017 edition)>Unit 1

Lesson 10: Infinite limits intro

# Infinite limits and asymptotes

Unbounded limits are represented graphically by vertical asymptotes and limits at infinity are represented graphically by horizontal asymptotes.

## Want to join the conversation?

• I thought you could never cross asymptotes, only approach; if so then you could definitely cross vertical asymptotes, like the multy-function 3y/(y^2-5)+sin(y)=x
• Here is the confusing thing about asymptotes. You can never cross a vertical asymptote, but you can cross a horizontal or oblique (slant) asymptote. The reason you cannot cross a vertical asymptote is that at the points on the asymptote, the function is undefined because the x value would make the denominator zero. I hope this makes sense!
• That wavy transformation using sin(x) was pretty cool.
• Can there be diagonal asymptotes?
• The horizontal cases seem a bit counter-intuitive for me. Why do we talk about it here. Are there any applications of the horizontal asymptotes in maths problems or in real life. I just felt a bit confused about graphs with horizontal asymptotes.
• Plenty of applications. A horizontal asymptote can often be interpreted as an upper or lower limit for a problem. For example, if we were to have a logistic function modeling the spread of the coronavirus, the upper horizontal asymptote (limit as x goes to positive infinity) would probably be the size of the Earth's population, since the maximum number of people that can get the virus is the number of people on the planet. If you are not familiar with the shape of a logistic graph please look it up to see.

Hyperbolic functions such as y = 1/x represent an inverse variation relationship between two quantities x and y. Many things in life vary inversely, for example the number of cigarettes someone smokes and their life expectancy. If you smoke 10 packs a day, your life expectancy will significantly decrease. The horizontal asymptote represents the idea that if you were to smoke more and more packs of cigarettes, your life expectancy would be decreasing. If it made sense to smoke infinite cigarettes, your life expectancy would be zero.
• why wont a vertical asymptote be crossed ? ( said in )
• a vertical asymptote could have a value for 1 point if you use a piecewise function, but it won't be crossed several times because that would result in several y-values for one x-value, which is not allowed for functions.
• How can we distinguish a horizontal and vertical asymptote in higher dimensions? Don't they loose meaning? Since everything is according to once point of reference (this might be the key, BUT) Does that mean that for every object in space we asign a coordinate plane to determine their point and asymptotes?

Sorry if I cannot get to explain myself or this may sound stupid, so many questions.
• We tend not to use the terms 'horizontal/vertical asymptote' in more than 2 dimensions. If a curve or surface was asymptotic to some line or plane, we would care more about finding the equation of that line or plane, which tells us everything we want to know about it, than about whether to call it horizontal or vertical.
(1 vote)
• i do not understand what are asymptotes. Is there nother video that explains it or can someone help ?

thank you
• An asymptote is when the line approaches an x or y value, but does not reach it.
To get a visual on this topic, I would plug the equation y=1/x into a graphing calculator. The asymptotes that you will see are x=0, (the line soars up to infinity on one side, and down to negative infinity on the other), and y=0, (as x goes to infinity, the line gets closer and closer to the x-axis, but it never touches).

I also tried to find a video on this topic, but I couldn't find one, so I hope my explanation helps you out.