Question

When I'm using limits to determine whether or not a function is continuous and it turns out it's removable discontinuity, what would the values look like?

Answer 1

For example, when it's jump discontinuity, the values of each one sided limit are different, but what would the limits of removable discontinuity look like? the same? finite?

Answer 2

The values would not hold a geometric ratio with the rest of the function. For instance, in a parabolic function that is discontinuous at point 2:
x = 0, y = 0^2 = 0
x = 1, y = 1^2 = 1
x = 1.5, y = 9/4
x = 1.999, y = 3.996001
x = 2, y = 0

Answer 3

I mean, when you are finding right hand and left hand limits, do they equal each other? Are they finite?

Answer 4

I still don't get it.....

Answer 5

It is a general limit, so yes. "Right hand" and "left hand" limits are equivalent.

Answer 6

but then how would I know if it's a discontinuity!? Or if it's removable!?

Answer 7

If the limit as x->c for f(x) is not equal to f(c).

Answer 8

so in other words, the left and right hand limits are equal, so the general limit exists, but it's not continuous because limit as x-->c for f(x) is not equal to f(c) and f(c)?

Answer 9

Or rather, let me put it more precisely. Let's say we have this function:
$$f(x) = \begin{cases}
x^2 & x \not = 2\\
0 & x = 2
\end{cases}$$
f(2) is not equal to 2^2, so the limit is not equal to what the function actually is. You are correct.