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: Understanding the Question:
The question provides a piecewise function and states that the limit of the function as \( x \) approaches 2 exists. For a limit to exist at a point, the left-hand limit (LHL) and the right-hand limit (RHL) must be equal.
Step 2: Key Formula or Approach:
For the limit to exist at \( x = 2 \):
\[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) \]
Step 3: Detailed Explanation:
1. Calculate the Left-Hand Limit (LHL):
\[ LHL = \lim_{x \to 2^-} (b - ax) = b - a(2) = b - 2a \]
2. Calculate the Right-Hand Limit (RHL):
\[ RHL = \lim_{x \to 2^+} (a + 2bx) = a + 2b(2) = a + 4b \]
3. Equate LHL and RHL:
\[ b - 2a = a + 4b \]
\[ -2a - a = 4b - b \]
\[ -3a = 3b \]
\[ \frac{a}{b} = \frac{3}{-3} = -1 \]
Step 4: Final Answer:
The value of \( \frac{a}{b} \) is -1.