Question

If the function 

is continuous at \(x = 0\), then \(\frac{1}{a} + \frac{1}{b} + \frac{4}{k}\) is equal to :} 
 

• A. \(-5\)
• B. \(5\)
• C. \(-4\)
• D. \(4\)

Hint:

Standard limits are much faster than L'Hôpital's rule for expressions involving logarithms and square roots. Always try to simplify the expressions first.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a function \(f(x)\) to be continuous at \(x = a\), the left-hand limit (LHL), the right-hand limit (RHL), and the value of the function at that point must be equal:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) \]
Step 2: Key Formula or Approach:
We will use standard limits:
1. \(\lim_{u \to 0} \frac{\ln(1+u)}{u} = 1\)
2. Taylor expansions or standard trigonometric limits: \(\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\)
Step 3: Detailed Explanation:
Part 1: Left-Hand Limit (LHL) at \(x = 0\)
\[ \text{LHL} = \lim_{x \to 0^-} \frac{1}{x} \log_e \left( \frac{1 + x/a}{1 - x/b} \right) = \lim_{x \to 0^-} \frac{\ln(1 + x/a) - \ln(1 - x/b)}{x} \]
Applying the property \(\lim_{x \to 0} \frac{\ln(1+cx)}{x} = c\):
\[ \text{LHL} = \frac{1}{a} - \left( -\frac{1}{b} \right) = \frac{1}{a} + \frac{1}{b} \]
Part 2: Right-Hand Limit (RHL) at \(x = 0\)
\[ \text{RHL} = \lim_{x \to 0^+} \frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1} = \lim_{x \to 0^+} \frac{\cos 2x - 1}{\sqrt{x^2 + 1} - 1} \]
Multiply by conjugate of the denominator:
\[ = \lim_{x \to 0^+} \frac{-(1 - \cos 2x)(\sqrt{x^2 + 1} + 1)}{(x^2 + 1) - 1} = \lim_{x \to 0^+} \frac{-(1 - \cos 2x)}{x^2} \cdot (\sqrt{x^2 + 1} + 1) \]
Since \(\lim_{x \to 0} \frac{1 - \cos 2x}{(2x)^2} = \frac{1}{2}\), then \(\lim_{x \to 0} \frac{1 - \cos 2x}{x^2} = \frac{1}{2} \cdot 4 = 2\):
\[ \text{RHL} = -2 \cdot (1 + 1) = -4 \]
Part 3: Equating for Continuity
For continuity at \(x = 0\), \(\text{LHL} = \text{RHL} = f(0)\):
\[ \frac{1}{a} + \frac{1}{b} = -4 = k \]
So, \(k = -4\) and \(\frac{1}{a} + \frac{1}{b} = -4\).
We need to find \(\frac{1}{a} + \frac{1}{b} + \frac{4}{k}\):
\[ \frac{1}{a} + \frac{1}{b} + \frac{4}{k} = -4 + \frac{4}{-4} = -4 - 1 = -5 \]
Step 4: Final Answer:
The value is \(-5\).