Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
Let $f(x) = \begin{cases} a & x = \frac{2}{3} \\ \frac{\sqrt{9x+3}-\sqrt{3}}{\sqrt{9-7x+4}} & x > \frac{2}{3} \end{cases}$ If $f(x)$ is continuous at $x = \frac{2}{3}$, then the value of $\frac{a}{b}$ is
Step-by-Step Solution
Key Concept: Apply rationalization techniques to evaluate limits of indeterminate forms and use continuity conditions.
We need to find $\lim_{x \to 2} f(x) = f(\frac{2}{3})$. Computing the limit: $\lim_{x \to 2} \frac{\sqrt{7x-5}}{\sqrt[3]{3x+1}-a} = a$. For the denominator to be zero, we need $b = 3$. After rationalization and simplification, we obtain $a = 0$.
Correct Answer: 3