To solve question 15, let's analyze the limit expression:

$\lim_{x \to 1} \frac{a \sqrt{x + 3} - b}{x - 1} = 1$

Our goal is to determine the value of $\frac{b}{a}$.

Step 1: Analyze the limit structure

To compute the limit, notice that the expression involves a form similar to $\frac{0}{0}$ as $x \to 1$. This suggests using L'Hôpital's Rule or algebraic manipulation.

Step 2: Rewrite and apply derivatives if needed

1. Substitute $x = 1$ to confirm it is an indeterminate form: $\text{Numerator: } a \sqrt{1 + 3} - b = a \cdot 2 - b = 2a - b$ $\text{Denominator: } 1 - 1 = 0$ Since both the numerator and denominator go to zero, we have a $\frac{0}{0}$ indeterminate form, so we can use L'Hôpital's Rule.

2. Differentiate the numerator and denominator with respect to $x$:

   • Numerator: $\frac{d}{dx} \left(a \sqrt{x + 3} - b\right) = \frac{a}{2\sqrt{x + 3}}$
   • Denominator: $\frac{d}{dx} (x - 1) = 1$

   Therefore, the limit becomes: $\lim_{x \to 1} \frac{a}{2\sqrt{x + 3}} = 1$

3. Substitute $x = 1$ into the simplified expression: $\frac{a}{2\sqrt{1 + 3}} = 1$ $\frac{a}{2 \cdot 2} = 1$ $\frac{a}{4} = 1$ Solving for $a$, we get: $a = 4$

Step 3: Substitute $a = 4$ to find $b$

Since $a \cdot 2 = b$,