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