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### Course: Precalculus>Unit 10

Lesson 2: Estimating limit values from graphs

# One-sided limits from graphs: asymptote

This video explores estimating one-sided limit values from graphs. As x approaches 6 from the left, the function becomes unbounded with an asymptote, making the left-sided limit nonexistent. However, when approaching 6 from the right, the function approaches -3, indicating that the right-handed limit exists. Sal's analysis highlights the importance of understanding limits from both sides.

## Want to join the conversation?

• Does this mean that the general limit of g(x) as x approaches 6 does not exist? In previous videos Sal stated that if both one-sided limits are not equal, the general limit does not exist. But is it meaningful in this case to say that the one-sided limits are not equal, when one of them does not exist?
• The general limit as x approaches 6 does not exist. The limit as x approaches from the left is positive infinity, and thus does not exist. The limit as x approaches 6 from the right is -3.
• My question may seem dull at first, however I couldn't find an answer to it.
Based on my understanding on limits the main function of a limit is to find the near approximated output value when the function is not defined at that output. But what if the function was defined at that point, will the limit become useless?
In other words, are limits useful only at undefined points?
• what are limits exactly used for ?
• Calculus limits have a wide range of applications in various fields. Engineers rely on limits for designing structures, analyzing circuits, optimizing systems, and solving differential equations. In finance, limits are used to calculate interest rates, evaluate investments, and assess risk and probability. They are also employed in pharmacokinetics to determine optimal dosing. A fun example is tracking aircraft: by collecting position data points over a short time interval and taking the limit as the interval approaches zero, we can calculate the aircraft's instantaneous velocity at any given moment. In summary, calculus limits are incredibly versatile and find applications in numerous areas, from engineering and finance to medicine and even more.
• So, the limit of g(x)as x →6 doesn't exist, right ?
• Correct, because the limit from the negative side is not equal to the limit from the positive side (and because the limit does not exist for the asymptote)
• Can I say its unbounded, instead of saying "it does not exist"?
• Yes, you can use the term "unbounded" to describe the behavior of the function g(x) as x approaches six from the left, instead of saying "it does not exist." Saying that the limit is unbounded means that the function grows without bound and becomes infinitely large as x approaches six from the left. Both "unbounded" and "does not exist" convey the same idea in this context.
• What kind of functions acutally produces this kind of "unbounded" limits?
• tan(x), 1/x, and any other function with a vertical asymptote, like many rational functions.
• I am still a little confused about the limit of g(x) as "x" approaches 6 from the left. How come that limit does not exist, and is not equal to infinity?
• Infinity is not a number, it is a concept. We use it to concisely say that we're approaching something but never being able to attain it, and are describing the manner in which it does so. As I get closer and closer to 6 from the left, I get higher and higher in y-value. There is no y-value such that x will actually "touch" 6 from the left, so the limit does not exist.
• I'm sorry but shouldn't Sal have answer the limit of f(x) as x---> 6 from the left side, shouldn't the answer be infinity? why did he write does not exist(I understand it doesn't because both sides are not equal, but if we look at it in more details technically it doesn't exist but from the left side it should be equal to infinity and from the right side is equal to -3
• correct. because he put the "-" next to the 6, it means that it is coming in from a value less than x=6. If he had not put it, then he would be correct because without a "+" or "-" then it means across the entire graph, not just a greater than or less than value of x=6.
(1 vote)
• What's the difference between an undefined limit and a limit that doesn't exist?