67. What kind of asymptotes does \( \left(x^{\wedge} 2-4\right) /(x- \) \( 5) \) have?

To determine the asymptotes of the function \( f(x) = \frac{x^{2} - 4}{x - 5} \), we'll examine both vertical and oblique (slant) asymptotes. ### 1. **Vertical Asymptote** A vertical asymptote occurs where the denominator of the function is zero (and the numerator is not zero at that point). **Find where the denominator is zero:** \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] **Check the numerator at \( x = 5 \):** \[ x^{2} - 4 = 5^{2} - 4 = 25 - 4 = 21 \neq 0 \] Since the numerator is not zero when \( x = 5 \), there is a vertical asymptote at: \[ x = 5 \] ### 2. **Oblique (Slant) Asymptote** An oblique asymptote occurs when the degree of the numerator is one higher than the degree of the denominator. In this case, the numerator is degree 2 and the denominator is degree 1, so an oblique asymptote exists. **Perform polynomial division of \( x^{2} - 4 \) by \( x - 5 \):** \[ \frac{x^{2} - 4}{x - 5} \] **Divide \( x^{2} \) by \( x \):** \[ x^{2} \div x = x \] Multiply \( x \) by \( x - 5 \): \[ x(x - 5) = x^{2} - 5x \] Subtract this from the original polynomial: \[ (x^{2} - 4) - (x^{2} - 5x) = 5x - 4 \] **Divide \( 5x \) by \( x \):** \[ 5x \div x = 5 \] Multiply \( 5 \) by \( x - 5 \): \[ 5(x - 5) = 5x - 25 \] Subtract this from the current polynomial: \[ (5x - 4) - (5x - 25) = 21 \] The division yields: \[ \frac{x^{2} - 4}{x - 5} = x + 5 + \frac{21}{x - 5} \] As \( x \) approaches infinity, the term \( \frac{21}{x - 5} \) approaches 0, so the oblique asymptote is: \[ y = x + 5 \] ### **Summary of Asymptotes:** - **Vertical Asymptote:** \( x = 5 \) - **Oblique Asymptote:** \( y = x + 5 \) **Graphical Representation:** - The function approaches the vertical line \( x = 5 \) but never crosses it. - As \( x \) becomes large (positive or negative), the function approaches the straight line \( y = x + 5 \). **Final Answer:** The graph has a vertical asymptote at x = 5 and an oblique (slant) asymptote given by y = x + 5.