Question

If \[ \lim_{x \to 9} f(x) = 6 \quad {and} \quad \lim_{x \to 9} g(x) = 3, \] then \[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \] is equal to

• A. \( 2 \)
• B. \( -2 \)
• C. \( \frac{1}{3} \)
• D. \( -\frac{1}{3} \)
• E. \( 0 \)

Hint:

To evaluate limits involving functions, apply the property: \[ \lim_{x \to a} \frac{f(x) - g(x)}{h(x)} = \frac{\lim_{x \to a} f(x) - \lim_{x \to a} g(x)}{\lim_{x \to a} h(x)} \] if the denominator is nonzero.

Answer 1

Answer: (E)

Evaluating the given limit: \[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \] \[ = \frac{\lim_{x \to 9} f(x) - 2 \lim_{x \to 9} g(x)}{\lim_{x \to 9} g(x)} \] \[ = \frac{6 - 2(3)}{3} = \frac{6 - 6}{3} = \frac{0}{3} = 0 \] Thus, the correct answer is (E).