Limits, Continuity & Differentiability
Standard Limits — Roots of Quadratic
nta_pyq_2024_apr
Grade 12
Question:
If $\alpha=\displaystyle\lim_{x\to0^+}\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}$ and $\beta=\displaystyle\lim_{x\to0}(1+\sin x)^{\frac{1}{2}\cot x}$ are the roots of the quadratic equation $ax^2+bx-\sqrt{e}=0$, then $12\log_e(a+b)$ is equal to
Step-by-Step Solution
Key Concept: $\alpha=\lim_{x\to0^+}e^{\sqrt{x}}\cdot\frac{e^{\sqrt{\tan x}-\sqrt{x}}-1}{\sqrt{\tan x}-\sqrt{x}}=e^0\cdot1=1$. For $\beta$: $\lim(1+\sin x)^{\cot x/2}=e^{\lim\frac{\sin x}{2\tan x}}=e^{1/2}$.
$\alpha=1$, $\beta=\sqrt{e}$. $a=-1$, $b=1+\sqrt{e}$. $12\ln(a+b)=12\times\frac{1}{2}=6$.
Correct Answer: 6