Questions: For what values of a does the limit as x approaches a of r(x) equal to r(a) if r is a rational function? Choose the correct answer below. A. Those values of a for which the denominator 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 numerator of the function r is zero. D. Those values of a for which the denominator of the function r is not zero.

Solution Steps

To determine for which values of \( a \) the limit \(\lim _{x \rightarrow a} r(x)=r(a)\) holds for a rational function \( r \), we need to consider the properties of rational functions. A rational function is defined as the ratio of two polynomials. The limit of a rational function as \( x \) approaches \( a \) will equal the function's value at \( a \) if the function is continuous at \( a \). Continuity of a rational function at \( a \) requires that the denominator is not zero at \( a \).

We are given a rational function \( r(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) is the numerator and \( q(x) \) is the denominator. For this example, we have: \[ p(x) = x^2 - 1 \] \[ q(x) = x - 1 \]

To determine if the limit \(\lim_{x \to a} r(x) = r(a)\) holds, we need to check if the function is continuous at \( x = a \). A rational function is continuous at \( x = a \) if the denominator \( q(x) \) is not zero at \( x = a \).

We evaluate the denominator at \( x = 1 \): \[ q(1) = 1 - 1 = 0 \]

Since the denominator is zero at \( x = 1 \), the function \( r(x) \) is not continuous at \( x = 1 \).

Final Answer