Find the vertical and horizontal asymptotes of the function \(f(x) = \frac{x^2 - 4}{x - 2}\).
Answer
Step 3: Find the horizontal asymptote. Since the degree of the numerator is equal to the degree of the denominator in the original function, the horizontal asymptote is found by dividing the leading coefficients, which gives \(y=1\).
Step 1 ：Step 1: Simplify the equation. We can factor the numerator to get \(f(x) = \frac{(x-2)(x+2)}{x-2}\). When \(x\neq2\), we can cancel the \(x-2\) terms to get \(f(x) = x + 2\).
Step 2 ：Step 2: Find the vertical asymptote. This is where the denominator of the original function equals zero, so \(x-2=0\) which gives \(x=2\).
Step 3 ：Step 3: Find the horizontal asymptote. Since the degree of the numerator is equal to the degree of the denominator in the original function, the horizontal asymptote is found by dividing the leading coefficients, which gives \(y=1\).
