Limits, Continuity & Differentiability
Continuity — Limit Evaluation
nta_pyq_2024_apr
Grade 12
Question:
If the function $f(x)=\begin{cases}\dfrac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x\neq0\\ a\log_e2\log_e3, & x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ is equal to:
Step-by-Step Solution
Key Concept: Factorise numerator: $72^x-9^x-8^x+1=(8^x-1)(9^x-1)$. Denominator: $\sqrt{2}-\sqrt{1+\cos x}=\frac{1-\cos x}{\sqrt{2}+\sqrt{1+\cos x}}$. Use standard limits $\frac{a^x-1}{x}\to\ln a$ and $\frac{1-\cos x}{x^2}\to\frac{1}{2}$.
$a=24\sqrt{2}$. $a^2=1152$.
Correct Answer: 2