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## Calculus, all content (2017 edition)

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

Lesson 7: Continuity at a point

# Discontinuity points challenge example

Based on out definition of continuity, we can see the relationship between points of discontinuity and two-sided limits. Created by Sal Khan.

## Want to join the conversation?

• I am just not understanding the concept of limits. why do we use them? what do limits mean? •  Limits have many uses. You could have an equation demonstrating the power a car's engine is able to put out in comparison to friction. The limit of this function,could easily be the maximum velocity the car could accomplish. You could have an equation relating the number of work hours logged by an employee in a day as compared to productivity. Because such a function would have a limited domain, it may be true that none of the local minimums and maximums you would find through derivative tests would be the overall min or max of that equation in that domain, thus you would need to compare them to the limit in order to maximize your efforts. Basically, limits have many uses, although they are most commonly used to test the maximum capability or productivity of people and machines, so as to get the best use out of them. Each one means something a little different depending on the context of the equation, but with a little critical thinking you can come up with that answer. I hope I've helped clarify this topic for you a little bit.
• I am just wondering whether the statement limit does not exist implies discontinuity? • By definition a limit exists if the limit from the right and the limit from the left approach the same value. Therefore, if a limit does not exist, either;
the left-handed limit and the right-handed limit approach two different numbers - which would be a jump discontinuity or
they approach infinity or negative infinity, implying an asymptote, which also causes a discontinuity.
• If lim f(x) (x → k) does not exist, do we have to check the point of discontinuity k? • Why is k equal to 8? Does it not represent the y-axis values? I would have thought that it would be equal to 7. • how to determine the value of x for which the the function is discontinuous and the only given is the function f(x) and there is no graph?
(1 vote) • There are three primary sources of discontinuity:
1. A point where a piecewise function changes and there is a sudden jump in value. For example: f(x) = 2x where x < 2, and 400x³ ≥ 2
is discontinuous as x = 2.
2. A point where the function is not defined or fails to exist (such as division by zero).
3. A point where the function switches from having a real value to a complex value.
• Okay, here's an odd case: What about 1/x? 1/x is not defined at 0, but the limit of 1/x as x -> 0 is ALSO not defined. Or, rather, it doesn't exist. Does this mean that 1/x qualifies as continuous, or are "function is not defined" and "limit does not exist" considered different things? My intuition says "1/x is not continuous" simply because, well, just look at it.

Is there ANY function that's undefined at a given point that's still considered continuous, or must a function be defined? • So, if both sided limits exist, then limit exist, but it is just not continuous at that point k..?
(1 vote) • Almost. The limit exists if both of the one-sided limits exist AND are exactly equal. The function itself does not have to equal what the limits equal or even be defined.

Remember that the limit of a function at a particular point IS NOT necessarily the same value as the function itself. In fact, the limit can exist where the function is undefined OR the function can exist but the limit fail to exist at that point.
• Do limits exists in our physical world?   