Question Number 183673 by universe last updated on 28/Dec/22
$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\boldsymbol{\mathrm{lim}}}\:\:\left[\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\frac{\boldsymbol{\pi}}{\mathrm{2}β\boldsymbol{{a}\mathrm{x}}}\right)\right]^{\boldsymbol{\mathrm{sec}}^{\mathrm{2}} \left(\frac{\boldsymbol{\pi}}{\mathrm{2}β\boldsymbol{\mathrm{bx}}}\right)} \\ $$
Answered by AlexandreT last updated on 28/Dec/22
$${lim}\:\left[{sen}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{2}β{ax}}\right)β\mathrm{1}\right]{sec}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{2}β{bx}}\right) \\ $$$${x}\rightarrow\mathrm{0} \\ $$$${sec}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{2}β{bx}}\right)=\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left(\mathrm{2}β{bx}\right)} \\ $$$$β{lim}\frac{\mathrm{1}β{sen}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{2}β{ax}}\right)}{\mathrm{1}β{sen}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{2}β{bx}}\right)} \\ $$$${x}\rightarrow\mathrm{0} \\ $$$$β{lim}\frac{\mathrm{1}β\left(\frac{\pi}{\mathrm{2}β{ax}}\right)^{\mathrm{2}} }{\mathrm{1}β\left(\frac{\pi}{\mathrm{2}β{bx}}\right)^{\mathrm{2}} } \\ $$$${x}\rightarrow\mathrm{0} \\ $$$$β{lim}\frac{\frac{\left(\mathrm{2}β{ax}\right)^{\mathrm{2}} β\pi^{\mathrm{2}} }{\left(\mathrm{2}β{ax}\right)^{\mathrm{2}} }}{\frac{\left(\mathrm{2}β{bx}\right)^{\mathrm{2}} β\pi^{\mathrm{2}} }{\left(\mathrm{2}β{bx}\right)^{\mathrm{2}} }} \\ $$$${x}\rightarrow\mathrm{0} \\ $$$$β{lim}\frac{\left(\mathrm{2}β{ax}\right)^{\mathrm{2}} β\pi^{\mathrm{2}} }{\left(\mathrm{2}β{ax}\right)^{\mathrm{2}} }Γ\frac{\left(\mathrm{2}β{bx}\right)^{\mathrm{2}} }{\left(\mathrm{2}β{bx}\right)^{\mathrm{2}} β\pi^{\mathrm{2}} } \\ $$$${x}\rightarrow\mathrm{0} \\ $$$$=β\left[\frac{\mathrm{2}^{\mathrm{2}} β\pi^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }Γ\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} β\pi^{\mathrm{2}} }\right] \\ $$$$=\:β\mathrm{1} \\ $$