Finding Positive Values of b for Continuous Function

[Loppyfoot]

Homework Statement

For what positive values of b is f continuous for all real numbers x?
f(x) = ((x-1)(x²-4))/(x²-b)

So I go one value of b for the function to be continuous. I got that b=4. How do I find any others? If there even are any others?

 

[LCKurtz]

I suppose you mean "a", not "b". And no, if a = 4 it isn't continuous for all values of x because then it is an undefined 0/0 form when x = ±2. And if you cancel the offending factors it is no longer the same function.

 

[Loppyfoot]

But when b=4, the top x^2-4 and the bottom x^2-4 cancel out to leave x-1.

 

[LCKurtz]

Yes, but they are no longer the same function since they don't have the same domain. For example consider the function y = x/x. This is not defined when x = 0, so its domain is x ≠ 0. But if you cancel the x's you have y = 1 which is defined for all x. The two functions do not have the same domain so they are not the same function.

 

[Loppyfoot]

So are there any values of b that make this function continuous?

 

[LCKurtz]

I think you know the answer. You would need a positive b (positive was given) so (x²-b) never gives you a zero in the denominator for any x, eh?

 

[Loppyfoot]

So there are no values of b that makes this function continuous?

 

[LCKurtz]

That makes it continuous for all x, which is what was required.

 

[Loppyfoot]

Alright, thanks LC!