Question

Define \[ f(x) = \begin{cases} b - ax, & \text{if } x < 2 \\ 3, & \text{if } x = 2 \\ a + 2bx, & \text{if } x > 2 \end{cases} \]

If \( \lim_{x \to 2} f(x) \) exists, then find \( \frac{a}{b} \).

• A. 1
• B. -1
• C. \(\frac{2}{3}\)
• D. \(\frac{3}{20}\)

Hint:

Limit existence doesn't require the function to be continuous. Here, LHL = RHL, but they don't necessarily equal $f(2)=3$.

Answer 1

The Correct Option is B

Step 1: Condition for Limit Existence
LHL (at $x=2$) = RHL (at $x=2$).

Step 2: Evaluate LHL
$LHL = \lim_{x \to 2^-} (b - ax) = b - 2a$.

Step 3: Evaluate RHL
$RHL = \lim_{x \to 2^+} (a + 2bx) = a + 4b$.

Step 4: Solve for $a/b$
$b - 2a = a + 4b \implies -3a = 3b \implies \frac{a}{b} = -1$.
Final Answer: (B)