Crazy Limits!!! 1

Calculus Level 5

Let

\[\displaystyle I = \lim_{x \to 0}\left(\frac{1}{x^{5}} \int_0^{x} e^{-t^{2}}dt -\frac{1}{x^{4}} + \frac{1}{3x^{2}} \right) \]

and

\[\displaystyle K= \lim_{n \to \infty}\left(\frac{[1^{2}\sqrt{7}] + [2^{2}\sqrt{7}]+ \ldots + [n^{2}\sqrt{7}]}{n^{3}} \right) .\]

Then find

\[\displaystyle \pi^{2} \times I + \frac{9K^{2}}{4}.\]

Details and Assumptions

  • \([x]\) denotes the floor function.

  • Use \(\pi^2 = 10\).

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