Question

If $\lim_{x \to 3} \left( \frac{x^2 - ax - 3a}{x - 3} \right) = 5$, then $a + b =$

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When a limit of a fraction $\frac{f(x)}{g(x)}$ exists finitely at $x=c$ and $g(c)=0$, it is a hard rule that $f(c)$ must also be $0$. Using this initial condition is often enough to solve for unknown parameters without fully evaluating the limit.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given a limit of a rational function that evaluates to a finite non-zero value ($5$) as $x$ approaches $3$. Since the denominator evaluates to zero ($3-3=0$) at $x=3$, the limit can only exist if the numerator also evaluates to zero at $x=3$. This creates an indeterminate form of type $\frac{0}{0}$.

Step 2: Key Formula or Approach:
1. Set the numerator expression to zero and substitute $x=3$: $f(3) = 3^2 - a(3) - 3b = 0$. 2. Simplify the resulting equation to find a relationship between $a$ and $b$.

Step 3: Detailed Explanation:
Let the given limit be $L = \lim_{x \to 3} \frac{x^2 - ax - 3b}{x - 3} = 5$. When we substitute $x = 3$ directly into the denominator, we get $3 - 3 = 0$. For the limit of the fraction to exist and be a finite number (like 5), the numerator must also approach zero as $x$ approaches 3. This avoids a $\frac{\text{non-zero}}{0}$ situation, which would lead to an undefined limit ($\pm\infty$). Therefore, we must have: \[ \lim_{x \to 3} (x^2 - ax - 3b) = 0 \] Substituting $x = 3$ into the numerator: \[ (3)^2 - a(3) - 3b = 0 \] \[ 9 - 3a - 3b = 0 \] Now, simplify this equation to find the value of $a+b$: Divide the entire equation by 3: \[ 3 - a - b = 0 \] Rearranging the terms yields: \[ a + b = 3 \]
Step 4: Final Answer:
The value of $a + b$ is 3.