Analisi matematica/Esempi derivata

Sviluppi di funzioni notevoli mediante la formula di Mac-Laurin

1. $e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\mathrm{....}+\frac{x^{n-1}}{(n-1)!}+\frac{x^n}{n!}{e}^{\theta x}$
2. $a^x=1+\frac{x\log a}{1!}+\frac{(x\log a{)}^2}{2!}+\mathrm{...}+\frac{(x\log a{)}^{n-1}}{(n-1)!}+\frac{(x\log a{)}^n}{n!}{a}^{\theta x}$
3. $senx=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}+\mathrm{...}+(-1{)}^k\frac{2^{2k+1}}{(2k+1)!}cos\theta x$
4. $cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\mathrm{...}+(-1{)}^k\frac{x^{2k}}{2k!}cos\theta x$

Calcolo del limite delle forme indeterminate

1. forma $\frac{0}{0}\underset{x\to -2}{lim}\frac{x^3+8}{x^5+32}=\frac{-8+8}{-32+32}=0$

$\underset{x\to -2}{lim}\frac{3x^2}{5x^4}=\frac{3(-2{)}^2}{5(-2{)}^4}=\frac{3}{20};$

$\underset{x\to 0}{lim}\frac{a^x-b^x}{c^x-d^x}=\frac{a^0-b^0}{c^0-d^0}=0$

$\underset{x\to 0}{lim}\frac{a^x\log a-b^x\log b}{c^x\log c-d^x\log d}=\frac{\log a-\log b}{\log c-\log d};$

$\underset{x\to 0}{lim}\frac{\log \frac{1-x^2}{1+x^2}}{sen{x}^2}=\frac{log1}{sen{0}^2}=\frac{0}{0}$

$\underset{x\to 0}{lim}\frac{\log \frac{1-x^2}{1+x^2}}{sen{x}^2}=\underset{x\to 0}{lim}\frac{d}{dx}(\frac{1-x^2}{1+x^2}):\frac{d}{dx}sen{x}^2=$

$\underset{x\to 0}{lim}\frac{\frac{-4x}{1-x^4}}{2senxcosx}=\underset{x\to 0}{lim}(-\frac{4}{2(1-x^4)cosx})\cdot \underset{x\to 0}{lim}\frac{x}{senx}=-2.$

1. forma $\frac{\mathrm{\infty }}{\mathrm{\infty }}\underset{x\to 0}{lim}\frac{\frac{1}{log(1-x)}}{cotgx}=\frac{\frac{1}{log1}}{cotg0}=\frac{\mathrm{\infty }}{\mathrm{\infty }}$