Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
If $f(x) = \begin{cases} \frac{e^{\sin x} - 1}{\sin x} & 0 < x < \frac{\pi}{6} \\ \lambda & x = 0 \end{cases}$ is continuous at $x = 0$, the value of $\frac{\ln(3)}{b^2}$ is equal to
Step-by-Step Solution
Key Concept: Continuity at a point requires the limit to exist and equal the function value at that point.
$f(x)$ is continuous at $x = 0$. Using the limit definition: $\lim_{x \to 0^-} \frac{(x^2 - 1)^{1/3} - (-1)}{x} = \lim_{x \to 0^+} \frac{\sqrt[3]{x^2 - 1} + 1}{x} = \lambda$. Evaluating $(\ln 2)^2 - 1.3 \cdot 1 = \lambda$ gives the relationship. The answer is 30.
Correct Answer: 30