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Formulario sui limiti notevoli





TAVOLA SUI LIMITI NOTEVOLI

\displaystyle\lim_{x\to 0} \frac{\sin x}{x} = 1 \displaystyle\lim_{x\to \pm\infty} {\left (1+\frac{1}{x} \right )}^x\! = e
\displaystyle\lim_{x\to 0} \frac{\sin{ax}}{bx} = \frac{a}{b} \displaystyle\lim_{x\to \pm\infty} {\left (1+\frac{a}{x} \right )}^{bx}\! = e^{ab}
\displaystyle\lim_{x\to 0} \frac{1-\cos{x}}{x} = 0 \displaystyle\lim_{x\to \pm\infty} {\left (\frac{x}{x+1} \right )}^x\! = \frac{1}{e}
\displaystyle\lim_{x\to 0} \frac{1-\cos(x)}{x^2} = \frac{1}{2} \displaystyle\lim_{x\to 0} {\left ( 1 + ax \right ) }^{\frac{1}{x}}\! = e^a
\displaystyle\lim_{x\to 0} \frac{\tan x}{x} = 1 \displaystyle\lim_{x\to 0} \frac{\log_a(1+x)}{x} = \log_a e = \frac{1}{\ln a}
\displaystyle\lim_{x\to 0}\frac{\arcsin x}{x} = 1 \displaystyle\lim_{x\to 0} \frac{\ln(1+x)}{x}\! = 1
\displaystyle\lim_{x\to 0} \frac{\arctan x}{x} = 1 \displaystyle\lim_{x\to 0} \frac{a^x-1}{x} = \ln a,\;\;a > 0
\displaystyle\lim_{x\to 1} \frac{(\arccos x)^2}{1-x} = 2  \displaystyle\lim_{x\to 0} \frac{e^x-1}{x} = 1
 \displaystyle\lim_{x\to 0} \frac{\sinh x}{x} = 1 \displaystyle\lim_{x\to 0} \frac{(1+x)^a-1}{x} = a
 \displaystyle\lim_{x\to 0} \frac{\tanh x}{x} = 1 \displaystyle\lim_{x\to 0} \frac{(1+x)^a-1}{ax} = 1

 

\displaystyle\lim_{f(x)\to 0} \frac{\sin f(x)}{x} = 1 \displaystyle\lim_{f(x)\to \pm\infty} {\left (1+\frac{1}{f(x)} \right )}^{f(x)}\! = e
\displaystyle\lim_{f(x)\to 0} \frac{\sin{af(x)}}{bf(x)} = \frac{a}{b} \displaystyle\lim_{f(x)\to \pm\infty} {\left (1+\frac{a}{f(x)} \right )}^{bf(x)}\! = e^{ab}
\displaystyle\lim_{f(x)\to 0} \frac{1-\cos{f(x)}}{f(x)} = 0 \displaystyle\lim_{f(x)\to \pm\infty} {\left (\frac{f(x)}{f(x)+1} \right )}^{f(x)}\! = \frac{1}{e}
\displaystyle\lim_{f(x)\to 0} \frac{1-\cos(f(x))}{f(x)^2} = \frac{1}{2} \displaystyle\lim_{f(x)\to 0} {\left ( 1 + af(x) \right ) }^{\frac{1}{f(x)}}\! = e^a
\displaystyle\lim_{f(x)\to 0} \frac{\tan f(x)}{f(x)} = 1 \displaystyle\lim_{f(x)\to 0} \frac{\log_a(1+f(x))}{f(x)} = \log_a e = \frac{1}{\ln a}
\displaystyle\lim_{f(x)\to 0}\frac{\arcsin f(x)}{f(x)} = 1 \displaystyle\lim_{f(x)\to 0} \frac{\ln(1+f(x))}{f(x)}\! = 1
\displaystyle\lim_{f(x)\to 0} \frac{\arctan f(x)}{f(x)} = 1 \displaystyle\lim_{f(x)\to 0} \frac{a^{f(x)}-1}{f(x)} = \ln a,\;\;a > 0
\displaystyle\lim_{f(x)\to 1} \frac{(\arccos f(x))^2}{1-f(x)} = 2  \displaystyle\lim_{f(x)\to 0} \frac{e^{f(x)}-1}{f(x)} = 1
 \displaystyle\lim_{f(x)\to 0} \frac{\sinh f(x)}{f(x)} = 1 \displaystyle\lim_{f(x)\to 0} \frac{(1+f(x))^a-1}{f(x)} = a
 \displaystyle\lim_{f(x)\to 0} \frac{\tanh f(x)}{f(x)} = 1 \displaystyle\lim_{f(x)\to 0} \frac{(1+f(x))^a-1}{af(x)} = 1