Limits, Continuity & Differentiability
Product of Decreasing Factors Approaching Zero
nta_pyq_2023_apr
Grade 12
Question:
$\displaystyle\lim_{n\to\infty}\left\{\left(2^{1/2}-2^{1/3}\right)\left(2^{1/2}-2^{1/5}\right)\cdots\left(2^{1/2}-2^{1/(2n+1)}\right)\right\}$ is equal to
1
0
$\sqrt{2}$
$\dfrac{1}{\sqrt{2}}$
Step-by-Step Solution
Key Concept: Each factor $2^{1/2}-2^{1/(2k+1)}<1$. The product is sandwiched between $(2^{1/2}-2^{1/3})^n\to0$ and $(2^{1/2}-2^{1/(2n+1)})^n\to0$ by squeeze theorem.
Limit $=0$.
Correct Answer: 2