Limiti notevoli

Principali

1. $$\underset{x\to 0}{\lim }\frac{\sin (ax)}{bx}\underset{\left[\frac{0}{0}\right]}{=}\frac{a}{b}$$
2. $$\underset{x\to \pm \infty }{\lim }{\left(1+\frac{a}{x}\right)}^{bx}\underset{[1^{\infty }]}{=}{e}^{ab}$$
3. $$\underset{x\to +\infty }{\lim }\frac{lo{g}_a(x^b)}{x^c}\underset{\left[\frac{\infty }{\infty }\right]}{=}0,c>0$$

Ricavabili

• Trigonometriche:

  • ${\lim }_{x\to 0}\frac{x}{\sin (x)}=1$
  • ${\lim }_{x\to 0}\frac{\tan (x)}{x}=1$
  • ${\lim }_{x\to 0}\frac{\arcsin (x)}{x}=1$
  • ${\lim }_{x\to 0}\frac{\arctan (x)}{x}=1$
  • ${\lim }_{x\to 0}\frac{1-\cos (x)}{x^2}=\frac{1}{2}$
  • ${\lim }_{x\to 0}\frac{\sin (ax)}{\sin (bx)}=\frac{a}{b}$

• Esponenziali e logaritmiche:

  • ${\lim }_{x\to 0}\frac{{\log }_a(1+bx)}{x}=\frac{b}{{\log }_e(a)}$
  • ${\lim }_{x\to 0}\frac{a^x-1}{x}={\log }_e(a)$
  • ${\lim }_{x\to 0}\frac{(1+x{)}^a-1}{x}=a$
  • ${\lim }_{x\to 0}\frac{e^{ax}-e^{bx}}{x}=a-b$
  • ${\lim }_{x\to 0^{+}}{x}^c\cdot lo{g}_a(x^b)=0$
  • ${\lim }_{x\to +\infty }\frac{a^x}{x^c}=+\infty ,a>1,c>0$