**How do I set these Calculus problems up?**

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*Give examples of functions which are discontinuous at some point, c, and explain why they are discontinuous. State whether the discontinuities are removable or not.*

a) No functional value f(c) does not exist

b) The limit of the function does not exist.

c) The limit of the function does not equal the functional value.

a) f(x) = (x^2 -1) / (x-1)

f(x) is discontinuous at x = 1 because this causes a division by zero.

It is removable however.

x^2 -1 = (x+1)(x-1)

Canceling the (x-1)’s you get f(x) = x+1

The limit as x goes to one of f(x) would be 1+1 = 2, but f(1) is not defined at x = 1. You would have an open circle at this point.

b) f(x) = 1/(x-1)

The function is discontinuous at x = 1.

This is not removable, and the function does not exist here.

c) You can use the example from a but change it just a little.

Let f(x) be the piece-wise function

f(x) = { (x^2 -1)/(x-1) if x ≠ 1

_____{ 5 if x = 1

The limit as x goes to 1 is 2 (as shown in example a), but f(1) = 5 (as defined by the piecewise function).

**40) The Fundamental Theorem of Calculus Explained**

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