Question

for what values of a does \\(\lim_{x \to a} r(x) = r(a)\\) if r is a rational function?

choose the correct answer below.

a. those values of a for which the numerator of the function r is zero.
b. those values of a for which the numerator of the function r is not zero.
c. those values of a for which the denominator of the function r is zero.
d. those values of a for which the denominator of the function r is not zero.

Explanation:

Define rational function continuity

A rational function \(r(x)\) is defined as the ratio of two polynomial functions:

$$r(x) = \frac{p(x)}{q(x)}$$

where \(p(x)\) and \(q(x)\) are polynomials.

State the limit condition

The equation \(\lim_{x \to a} r(x) = r(a)\) is the definition of continuity at \(x = a\).
For this equation to hold, the function \(r(x)\) must be defined at \(x = a\).

Determine the domain restriction

The rational function \(r(a) = \frac{p(a)}{q(a)}\) is defined if and only if the denominator is not zero:

$$q(a) eq 0$$

Match with the given choices

If the denominator \(q(a)
eq 0\), then \(r(x)\) is continuous at \(a\), meaning \(\lim_{x \to a} r(x) = r(a)\).
This corresponds to those values of \(a\) for which the denominator of the function \(r\) is not zero.

Answer:

• A. Those values of a for which the numerator of the function r is zero.
• B. Those values of a for which the numerator of the function r is not zero.
• C. Those values of a for which the denominator of the function r is zero.
• D. Those values of a for which the denominator of the function r is not zero. (Correct answer)