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I'm trying to solve improper integrals but some patterns i can't do anything... This question is patterns of them. please help to to solve this question

It's okay that helping just one of them. I really need somebody's help...

questions is "State whether the sequence converges as $n\to\infty$; if it does, find the limit."

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Simon Fraser
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daysix
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1 Answers1

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Using, for example, this question we can see that $$\lim_{n\to\infty}\int_{-1/n}^{1/n}(\sin{(x^2)}-\cos{(2x)})\mathrm{d}x=\int_0^0(\sin{(x^2)}-\cos{(2x)})\mathrm{d}x=0$$ $$\lim_{n\to\infty}\int_{-1+1/n}^{1-1/n}\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\int_{-1}^1\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\pi$$

Peter Foreman
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