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f-x-2-x-0-x-t-2-1-t-2-dt-calculate-lim-x-0-f-x-




Question Number 146497 by mathmax by abdo last updated on 13/Jul/21
f(x)=(2/x)∫_0 ^x   (t^2 /( (√(1+t^2 ))))dt  calculate lim_(x→0) f(x)
$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2}}{\mathrm{x}}\int_{\mathrm{0}} ^{\mathrm{x}} \:\:\frac{\mathrm{t}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\mathrm{dt}\:\:\mathrm{calculate}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \mathrm{f}\left(\mathrm{x}\right) \\ $$
Answered by gsk2684 last updated on 13/Jul/21
lim_(x→0) ((2∫_0 ^x (t^2 /( (√(1+t^2 ))))dt)/x) apply  L′Hopitals rule  2lim_(x→0) ((((x^2 /( (√(1+x^2 ))))))/1)=0
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}\underset{\mathrm{0}} {\overset{{x}} {\int}}\frac{{t}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{dt}}{{x}}\:{apply}\:\:{L}'{Hopitals}\:{rule} \\ $$$$\mathrm{2}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\frac{{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\right)}{\mathrm{1}}=\mathrm{0} \\ $$

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