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### Course: Differential Calculus>Unit 1

Lesson 10: Types of discontinuities

# Types of discontinuities

This video discusses three discontinuity types: point/removable, jump, and asymptotic/infinite. Point/removable occurs when the two-sided limit exists but differs from the function's value. Jump arises when one-sided limits aren't equal, and asymptotic/infinite happens when the two-sided limit is unbounded. Continuity requires the two-sided limit to match the function's value.

## Want to join the conversation?

• Is an asymptotic discontinuity any different than an infinite discontinuity?
• They are the same thing – if you look in the `About` text it actually says "Asymptotic/infinite discontinuity".
• I understand that classification of discontinuities is 3 types
i. point removable
ii. jump discontinuities
iii.asympotic discontinuities

anything also
• There are also oscillating discontinuities. Look at the graph of f(x)=sin(1/x). It has no value or limit at x=0.
• I can't understand why the value of the y=x^2 graph at x=3 is 4, and not 9. Probably an obvious answer, but it's eluding me!
• So can you see the dot that is separated from the curve?
• Hi, I am learning how to evaluate functions by direct substitution right now. I was wondering why simply substituting or re-arranging a function would automatically give us the limit at that point. For example, in the two graphs on the left in this video, the y-value is defined at the x-value but the limit either doesn't equal that same y-value or doesn't exist. I want to see the actual functions that could result in these two graphs to better understand why we can directly substitute without fear of scenarios like these two. Is it only possible for piece-wise functions to create these types of scenarios? I feel like I am overlooking something and would really appreciate the help. Thanks in advance!
• A function can be determined by direct substitution if and only if lim_(x->c)_ f(x) = f(c). In other words, as long as the function is not discontinuous, you can find the limit by direct substitution.

There is also another way to find the limit at another point, and that is by looking for a determinant for the indeterminate form by using other methods and defining it by using another function. For example, lim_(x->2) (x^2 + 4 x - 12)/(x - 2), determined directly, equals (0/0), indeterminant form. However, there are many ways to determine a function by simply simplifying the function when direct substitution yields the indeterminant form. For this example, you could simply factor the limit to get lim_(x->2)_ (x+6), x ≠ 2. The constraint is added to be mathematically correct when it comes to being equivalent to the limit beforehand. However, say you found a function that is similar to the simplified function, only without the constraint, called g(x) = (x+6). You can define that as your new limit: lim_(x->2) g(x) = 8, thus lim_(x->2) (x^2 + 4 x - 12)/(x - 2) = 8.

• Is there any difference between saying that a limit doesn't exist and saying that a limit is unbounded? I've only ever heard Sal saying a limit doesn't exist/there is no limit when a limit is being taken from both sides. This is the first time I can remember Sal saying something similar regarding a limit being taken from one side, however he called the limit unbounded this time.

I hope I explained myself clearly and if anyone could answer this question, that would be appreciated, thanks!
• Is a quadractic formula discontinous
• Well, the quadratic formula is a formula, so it can't be graphed.

However, a function related to the quadratic formula, a quadratic polynomial, is continuous over its entire interval.
• What would be an example of a function with both a removable and a non-removable discontinuity?
• What about the function which has one sided limit? For example if lim_(x->0) when approaching from the right exists, but lim_(x->0) when approaching from the left is asymptote ?
Does this qualify as asymptotic discontinuity or some kind of mix of jump/asymptotic discontinuity?