Calcular el límite: \( f(x)=\displaystyle\lim_{x \to{0}}{\displaystyle\frac{x^5+sen^2(2x^2)}{ln(1+x^3).arcsen(5x)}} \)
Usando conocidos infinitésimos equivalentes,
\( \displaystyle\lim_{x \to{0}}{\displaystyle\frac{x^5+\sin^2(2x^2)}{\ln(1+x^3)\cdot \arcsin(5x)}}=\displaystyle\lim_{x \to{0}}{\displaystyle\frac{x^5}{\ln(1+x^3)\cdot \arcsin(5x)}}+\displaystyle\lim_{x \to{0}}{\displaystyle\frac{\sin^2(2x^2)}{\ln(1+x^3)\cdot \arcsin(5x)}} \)
\( =\displaystyle\lim_{x \to{0}}{\displaystyle\frac{x^5}{x^3\cdot (5x)}}+\displaystyle\lim_{x \to{0}}{\displaystyle\frac{4x^4}{x^3\cdot (5x)}}=0+\dfrac{4}{5}=\dfrac{4}{5}. \)