For what values of a does \( \lim _{x \rightarrow a} r(x)=r(a) \) 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 not zero. B. Those values of a for which the denominator of the function \( r \) is zero. C. Those values of a for which the denominator of the function \( r \) is not zero. D. Those values of a for which the numerator of the function \( r \) is zero.

To determine the values of \( a \) for which \( \lim_{x \rightarrow a} r(x) = r(a) \) holds true for a rational function \( r(x) \), we need to consider the behavior of the function at the point \( a \). A rational function is defined as the ratio of two polynomials, say \( r(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) is the numerator and \( q(x) \) is the denominator. 1. **If \( q(a) \neq 0 \)**: The function \( r(a) \) is defined, and since rational functions are continuous at points where they are defined, we have \( \lim_{x \rightarrow a} r(x) = r(a) \). 2. **If \( q(a) = 0 \)**: The function \( r(a) \) is undefined, and thus the limit cannot equal \( r(a) \). Therefore, the limit \( \lim_{x \rightarrow a} r(x) = r(a) \) holds true if and only if the denominator \( q(a) \) is not zero. Thus, the correct answer is: **C. Those values of a for which the denominator of the function \( r \) is not zero.**