Question

Calcule

a)$\lim_{x \to+ \infty} \frac{e^{2x} }{x^{3} }$


b)$\lim_{x \to\ 0} \frac{x-tg\ x}{x^3}$

Answer 1

a)

$\lim_{x \to+ \infty} \frac{e^{2x} }{x^{3} } = \\ \lim_{x \to+ \infty} \frac{2e^{2x} }{3x^{2} }$

$\lim_{x \to+ \infty} \frac{4e^{2x} }{6x } \\ = \lim_{x \to+ \infty} \frac{8e^{2x} }{6 } = ∞$

b)

$\lim_{x \to\ 0} \frac{x-tg\ x}{x^3} \\ = \lim_{x \to\ 0} \frac{1- { \sec }^{2}x }{3 {x}^{2} }$

$= \lim_{x \to\ 0} \frac{-tg ^{2}x }{3 {x}^{2} } = \\ - \frac{1}{3} \lim_{x \to\ 0} {( \frac{ \tan(x) }{x}) }^{2} = - \frac{1}{3}$