Limits, Continuity & Differentiability
Limits
nta_abhyas_2025
Grade 12
Question:
If $\lim_{x \to \infty} \left(1 + p^2 + q^2(x)\right) = e^p$, then
p = -\frac{1}{2}, q \in \mathbb{R}
q = -\frac{1}{2}, p \in \mathbb{R}
p = -\frac{1}{2}, p \in \mathbb{R}
q = -\frac{1}{2}, p = 0
Step-by-Step Solution
Key Concept: For limits of the form $1^\infty$, convert to the exponential form and use $\lim f(x)^{g(x)} = e^{\lim g(x)\ln f(x)}$
Given that the limit is of the form $1^\infty$: $\lim_{x \to \infty} \left(\frac{p+qx^2}{x^2}\right)^{1/(x^2)}$. We use $e^\alpha = 5$ which gives $\lim_{x \to \infty} \left(\frac{px+qx}{x^2}\right) = 5$. Therefore $\lim_{(p+qx)}(1) = 5$, which means $p + q = 5$ where $p = 5$ and $q \in \mathbb{R}$. Thus $p - 5q = 5$.
Correct Answer: 5