Limits, Continuity & Differentiability
Limits
nta_abhyas_2025
Grade 12
Question:
Find $\lim_{x \to \infty} \left( \frac{p^{1/x} + q^{1/x} + r^{1/x}}{3} \right)^h$ [where $p, q, r, a > 0]$
Step-by-Step Solution
Key Concept: For $1^\infty$ indeterminate forms, use logarithmic differentiation and the approximation $e^z \approx 1 + z$ for small $z$.
Put $\frac{1}{x} = z$, so that $x \to \infty$ means $z \to 0$. The expression becomes $\lim_{z \to 0} \left( \frac{p^z + q^z + r^z}{3} \right)^{a/z}$, which is in the form $1^\infty$. Taking logarithm and using $e^z \approx 1 + z$ for small $z$: $\lim_{z \to 0} \frac{a}{z} \ln\left(\frac{p^z + q^z + r^z}{3}\right) = \frac{a}{3}\ln(pqr)$, so the limit is $(pqr)^{a/3}$.
Correct Answer: (pqr)^(a/3)