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lim-x-0-x-sin-t-dx-x-




Question Number 213055 by MrGaster last updated on 29/Oct/24
                  lim_(x→+∞) ((∫_0 ^x ∣sin t∣dx)/x)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\frac{\int_{\mathrm{0}} ^{{x}} \mid\mathrm{sin}\:{t}\mid{dx}}{{x}} \\ $$$$ \\ $$
Answered by Ghisom last updated on 29/Oct/24
∫_0 ^x ∣sin t∣dx=∣sin t∣∫_0 ^x dx=x∣sin t∣  lim_(x→∞)  ((x∣sin t∣)/x) =∣sin t∣
$$\underset{\mathrm{0}} {\overset{{x}} {\int}}\mid\mathrm{sin}\:{t}\mid{dx}=\mid\mathrm{sin}\:{t}\mid\underset{\mathrm{0}} {\overset{{x}} {\int}}{dx}={x}\mid\mathrm{sin}\:{t}\mid \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}\mid\mathrm{sin}\:{t}\mid}{{x}}\:=\mid\mathrm{sin}\:{t}\mid \\ $$

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