Question

Let \(\lim_{x \to a} f(x)g(x) = 16\) and \(\lim_{x \to a} \frac{f(x)}{g(x)} = 4\). If both \(\lim_{x \to a} f(x)\) and \(\lim_{x \to a} g(x)\) exist, then \(\lim_{x \to a} [f(x)+g(x)]\) is

• A. $\pm 10$
• B. $-16$
• C. $\pm 2$
• D. $16$
• E. $\pm 4$

Hint:

When both product and ratio are given, convert limits into algebraic equations.

Answer 1

The Correct Option is A

Concept: Let: \[ \lim_{x \to a} f(x) = L,\quad \lim_{x \to a} g(x) = M \] Then: \[ LM = 16,\quad \frac{L}{M} = 4 \]

Step 1: Form equations
\[ L = 4M \]

Step 2: Substitute into product
\[ (4M)(M) = 16 \Rightarrow 4M^2 = 16 \Rightarrow M^2 = 4 \Rightarrow M = \pm 2 \]

Step 3: Find $L$
\[ L = 4M = \pm 8 \]

Step 4: Compute sum
\[ L + M = \pm 8 \pm 2 = \pm 10 \] Final Conclusion:
Option (A)