# Calculus Problems Explained

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

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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