Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
Consider the function $f(x) = \begin{cases} \sqrt{x^2 - 9} & \text{for some condition} \end{cases}$, then
Step-by-Step Solution
Key Concept: Power functions $x^p$ are continuous at 0 for $p > -1$ and differentiable at 0 for $p > 0$.
Continuity at $x = 0$: $\lim_{h \to 0^-} (-h)^p \cos^{-1} 1 = 0$ if $p > -1$. Also $\lim_{h \to 0^+} (h)^p \cos^{-1} 1 = 0$ if $p > -1$. For differentiability at $x = 0$: $f'(0^-) = \lim_{h \to 0^-} \frac{(-h)^p \cos^{-1} 1}{h}$ and $f'(0^+) = \lim_{h \to 0^+} \frac{(h)^p \cos^{-1} 1}{h}$. Both limits equal 0 when $p > 0$.
Correct Answer: 1