Thread: continuous function

k(x) is a piecewise function

if k(x)= (x^2-2b+b^2)/(x-b) if x does not equal a
= 5 if x=a

then which of the following is true about k(x):
1) limx->a k(x) exists
2) k(x) is continuous at x=a
3) k(a) exists

so far i think that k(a) will exist because it will equal 5 right? realli not sure about the other two.

Hi chukie! First, let me make your notation easier to read:

Originally Posted by chukie

is a piecewise function



then which of the following is true about :

1)

2)

3)

so far i think that will exist because it will equal 5 right? realli not sure about the other two.

You are correct that is defined at .

For the others:

1. When taking a limit, we only care about what happens to a function as it approaches a certain value, not what happens at that value. So, the limit of as will be the same as the limit of the first "piece."

2. For a function to be continuous at a point , three things must be true:

*

*

*

You should know the first 2 by answering the other two parts of this problem. So, take the limit of as . Then, if the value of this limit is , then is continuous at , and it will be discontinuous otherwise.

You should be able to find which value for makes the function continuous.

thanks so much! ur awesome