For $a, b > 0$, let $ f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0, \\ \frac{x}{3}, & x = 0, \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a x \sqrt{x}}}, & x > 0 \end{cases} $ be a continuous function at $x = 0$. Then $\frac{b}{a}$ is equal to:
- 5
- 4
- 8
- 6
The Correct Option is D
Approach Solution - 1
To ensure continuity of \( f(x) \) at \( x = 0 \), the following condition must hold:
\(\lim_{x \to 0} f(x) = f(0) = 3.\)
Step 1: Evaluating the right-hand limit
We consider:
\(\lim_{x \to 0^+} \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b \sqrt{ax} \sqrt{x}} = 3.\)
Step 2: Rationalizing the numerator
Multiply the numerator and denominator by the conjugate of the numerator:
\(\lim_{x \to 0^+}
\frac{(\sqrt{ax + b^2x^2} - \sqrt{ax})(\sqrt{ax + b^2x^2} + \sqrt{ax})}
{b \sqrt{ax} \sqrt{x} (\sqrt{ax + b^2x^2} + \sqrt{ax})}.\)
Step 3: Simplifying the expression
This becomes:
\(\lim_{x \to 0^+}
\frac{ax + b^2x^2 - ax}{b x^{3/2} (\sqrt{ax + b^2x^2} + \sqrt{ax})}.\)
Step 4: Canceling and simplifying further
\(\lim_{x \to 0^+}
\frac{b^2}{b \sqrt{a} (\sqrt{a + b^2x} + \sqrt{a})}.\)
Step 5: Substituting \( x = 0 \)
\(\frac{b}{\sqrt{a} \cdot 2\sqrt{a}} = \frac{b}{2a}.\)
Step 6: Applying the continuity condition
Since the limit must equal \( f(0) = 3 \), we have:
\(\frac{b}{2a} = 3 \implies \frac{b}{a} = 6.\)
Approach Solution -2
For \( f(x) \) to be continuous at \( x = 0 \), we require:
\(\lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x)\).
Given: \(f(0) = \frac{0}{3} = 0\).
Right-hand limit:
Consider:
\(\lim_{x \to 0^+} \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b \sqrt{ax}}\).
To simplify, multiply the numerator and the denominator by the conjugate of the numerator:
\(\lim_{x \to 0^+} \frac{(\sqrt{ax + b^2x^2} - \sqrt{ax}) \times (\sqrt{ax + b^2x^2} + \sqrt{ax})}{b \sqrt{ax} \times (\sqrt{ax + b^2x^2} + \sqrt{ax})}\).
This simplifies to:
\(\lim_{x \to 0^+} \frac{ax + b^2x^2 - ax}{b \sqrt{ax} \times (\sqrt{ax + b^2x^2} + \sqrt{ax})}\).
Simplifying further:
\(\lim_{x \to 0^+} \frac{b^2x^2}{b \sqrt{ax} \times (\sqrt{ax + b^2x^2} + \sqrt{ax})}\).
Canceling terms and simplifying:
\(\lim_{x \to 0^+} \frac{bx}{\sqrt{a} \times 2 \sqrt{ax}} = \frac{b}{2a}\).
For continuity, we equate this limit to the value of \( f(0) \):
\(\frac{b}{2a} = 3 \implies \frac{b}{a} = 6\).
Therefore:
\(\boxed{6}\).
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