Rational Continuity

[JSmith]

A rational function

is continuous at x = a if what condition is satisfied?

Option 1: If g(a) is not equal to zero, Option 2: or if g(a) is defined. Those are two of my options, the others involve f(x), which theoretically could be undefined or equal to zero, correct?

 

[pka]

A rational function

is continuous at x = a if what condition is satisfied?

Option 1: If g(a) is not equal to zero, Option 2: or if g(a) is defined. Those are two of my options, the others involve f(x), which theoretically could be undefined or equal to zero, correct?

I don't know where your options come from, BUT
each of \(\displaystyle f~\&~g\) must be continuous at \(\displaystyle x=a\) and \(\displaystyle g\ne 0\).

 

[daon2]

g(x) is automatically defined at a, it is a polynomial.

 

[HallsofIvy]

Note that if g(a)= 0, even if \(\displaystyle \lim_{x\to a}\frac{f(x)}{g(x)}\) exists, as, for example, \(\displaystyle \lim{x\to 2}\frac{x^2- 4}{x- a}= \lim_{x\to 2} x+ 2= 4\), the function, \(\displaystyle \frac{f(x)}{g(x)}\) is NOT continuous at x= a because it is not defined there.