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Let-s-R-z-define-as-R-z-pi-0-z-f-2-t-dt-2pi-0-z-f-t-1-f-1-t-2-dt-and-both-integral-0-f-2-t-dt-0-f-t-1-f-1-t-2-dt-lim-z-

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Question Number 214511 by issac last updated on 11/Dec/24
Let′s R(z) define as R(z)=((π ∫_0 ^( z) f^2 (t)dt)/(2π ∫_0 ^( z) f(t)(√(1+(f^((1)) (t))^2 ))dt)) and both integral ∫_0 ^( ∞) f^( 2) (t)dt , ∫_0 ^( ∞) f(t)(√(1+(f^((1)) (t))^2 ))dt =∞ lim_(z→∞) ((π ∫_0 ^( z) f^( 2) (t)dt)/(2π ∫_0 ^( z) f(t)(√(1+(f^((1)) (t))^2 ))dt)) =lim_(z→∞) ((π f(z))/(2π (√(1+(f^((1)) (z))^2 )))) ..??
$$\mathrm{Let}'\mathrm{s}\:{R}\left({z}\right)\:\mathrm{define}\:\mathrm{as}\: \\ $$$${R}\left({z}\right)=\frac{\pi\:\int_{\mathrm{0}} ^{\:{z}} \:{f}^{\mathrm{2}} \left({t}\right)\mathrm{d}{t}}{\mathrm{2}\pi\:\int_{\mathrm{0}} ^{\:{z}} \:{f}\left({t}\right)\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({t}\right)\right)^{\mathrm{2}} }\mathrm{d}{t}} \\ $$$$\mathrm{and}\:\mathrm{both}\:\mathrm{integral} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:{f}^{\:\mathrm{2}} \left({t}\right)\mathrm{d}{t}\:,\:\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({t}\right)\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({t}\right)\right)^{\mathrm{2}} }\mathrm{d}{t}\:=\infty \\ $$$$\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi\:\int_{\mathrm{0}} ^{\:{z}} {f}^{\:\mathrm{2}} \left({t}\right)\mathrm{d}{t}}{\mathrm{2}\pi\:\int_{\mathrm{0}} ^{\:{z}} \:{f}\left({t}\right)\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({t}\right)\right)^{\mathrm{2}} }\mathrm{d}{t}} \\ $$$$=\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi\:{f}\left({z}\right)}{\mathrm{2}\pi\:\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({z}\right)\right)^{\mathrm{2}} }}\:..?? \\ $$

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