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Question: If f(n) = \(\lim _ { x \rightarrow 0 }\) \(\sum_{k = 1}^{n}\frac{e^{k^{3}x} - 1}{x}\), then \(\lim _...

If f(n) = lim⁡x→0\lim _ { x \rightarrow 0 } ∑k=1nek3x−1x\sum_{k = 1}^{n}\frac{e^{k^{3}x} - 1}{x}, then lim⁡x→∞\lim _ { x \rightarrow \infty } f(x)g(x)\frac{f(x)}{g(x)} is

A

Does not exist

B

2

C

0

D

Data inadequate

Answer

2

Explanation

Solution

f(n) = n2(n2+1)2\frac { \mathrm { n } ^ { 2 } \left( \mathrm { n } ^ { 2 } + 1 \right) } { 2 } , g(n) = (n(n+1)2)2\left( \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } \right) ^ { 2 }

⇒ lim⁡x→∞\lim _ { x \rightarrow \infty } x2(x2+1)42(x+1)2x2\frac { x ^ { 2 } \left( x ^ { 2 } + 1 \right) 4 } { 2 ( x + 1 ) ^ { 2 } x ^ { 2 } } = 2