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lim-x-0-0-x-2-sin-t-dt-x-3-

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Question Number 223304 by Nadirhashim last updated on 21/Jul/25
lim_(x→0) ((∫_0 ^x^2 sin((√t))dt )/x^3 ) =...?
$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{{x}^{\mathrm{2}} } {\int}}\boldsymbol{{sin}}\left(\sqrt{\boldsymbol{{t}}}\right)\boldsymbol{{dt}}\:}{\boldsymbol{{x}}^{\mathrm{3}} }\:=…? \\ $$
Answered by Raphael254 last updated on 21/Jul/25
f′(w) = (f(u(x)))′ = f′(u(x))×u′(x) = f′(u)×u′ = f′(u) du ∫_a ^( u) f′(w) = f′(u) du ∫_0 ^( x^2 ) sin((√t)) dt u = x^2 ⇒ du = 2x (d/dx)(∫_0 ^( x^2 ) sin((√t))×dt) = sin((√x^2 ))×2x = 2x×sin ∣x∣ lim_(x→0) ((2x×sin ∣x∣)/x^3 ) = 2lim_(x→0) ((sin ∣x∣)/x^2 ) sin ∣x∣ = { ((sin (x), if x ≥ 0)),((sin (−x), if x < 0)) :} if x ≥ 0: 2lim ((sin (x))/x^2 ); applying 
$$ \\ $$$${f}'\left({w}\right)\:=\:\left({f}\left({u}\left({x}\right)\right)\right)'\:=\:{f}'\left({u}\left({x}\right)\right)×{u}'\left({x}\right)\:=\:{f}'\left({u}\right)×{u}'\:=\:{f}'\left({u}\right)\:{du} \\ $$$$ \\ $$$$\int_{{a}} ^{\:{u}} \:{f}'\left({w}\right)\:=\:{f}'\left({u}\right)\:{du} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\:{x}^{\mathrm{2}} } {sin}\left(\sqrt{{t}}\right)\:{dt} \\ $$$$ \\ $$$${u}\:=\:{x}^{\mathrm{2}} \:\Rightarrow\:{du}\:=\:\mathrm{2}{x} \\ $$$$ \\ $$$$\frac{{d}}{{dx}}\left(\int_{\mathrm{0}} ^{\:{x}^{\mathrm{2}} } {sin}\left(\sqrt{{t}}\right)×{dt}\right)\:=\:{sin}\left(\sqrt{{x}^{\mathrm{2}} }\right)×\mathrm{2}{x}\:=\:\mathrm{2}{x}×{sin}\:\mid{x}\mid \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{x}×{sin}\:\mid{x}\mid}{{x}^{\mathrm{3}} }\:=\:\mathrm{2}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{sin}\:\mid{x}\mid}{{x}^{\mathrm{2}} } \\ $$$$ \\ $$$${sin}\:\mid{x}\mid\:=\:\begin{cases}{{sin}\:\left({x}\right),\:{if}\:{x}\:\geq\:\mathrm{0}}\\{{sin}\:\left(−{x}\right),\:{if}\:{x}\:<\:\mathrm{0}}\end{cases} \\ $$$$ \\ $$$${if}\:{x}\:\geq\:\mathrm{0}: \\ $$$$ \\ $$$$\mathrm{2lim}\:\frac{{sin}\:\left({x}\right)}{{x}^{\mathrm{2}} };\:{applying}\:\:\underbrace{\nleqq} \\ $$
Answered by Mathspace last updated on 25/Jul/25
=lim_(x→0) ((2xsin(∣x∣))/(3x^2 )) (hospital) =lim_(x→0) (2/3)×((sin∣x∣)/x) =(2/3)si x→0^+ et−(2/3)si x→0^−
$$={lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{2}{xsin}\left(\mid{x}\mid\right)}{\mathrm{3}{x}^{\mathrm{2}} }\:\:\:\left({hospital}\right) \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{2}}{\mathrm{3}}×\frac{{sin}\mid{x}\mid}{{x}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}{si}\:{x}\rightarrow\mathrm{0}^{+} \:{et}−\frac{\mathrm{2}}{\mathrm{3}}{si}\:{x}\rightarrow\mathrm{0}^{−} \\ $$
Commented by Raphael254 last updated on 26/Jul/25
you didn′t derivate 2xsin(∣x∣)
$${you}\:{didn}'{t}\:{derivate}\:\mathrm{2}{xsin}\left(\mid{x}\mid\right) \\ $$

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