Detalji
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\(\color{green}{1.\:\:Limes\:elementarnih\:funkcija}:\:\) Za sve elementarne funkcije u svakoj točki \(a\) u kojoj su definirane vrijedi

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }f\left( x\right) =f\left( a\right) \tag{L1}}}\]

\(\color{green}{2.\:\:Limes\:konstantne\:funkcije}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }c =c \tag{L2}}}\]

\(\color{green}{3.\:\:Limes\:identične\:funkcije}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }x =a \tag{L3}}}\]

\(\color{green}{4.\:\:Limes\:potencije}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }x^{n}=a^{n} \tag{L4}}}\]

\(\color{green}{5.\:\:Limes\:korijena}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }\sqrt[n]{x}=\sqrt[n]{a} \tag{L5}}}\]

Limes i algebarske operacije

Neka su \(f\) i \(g\) bilo koje dvije funkcije koje imaju limes u točki \(a\). Uvedimo oznake:

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }f\left( x\right) =L,~\ \ \ \ \ \ \underset{%
x\rightarrow a}{\lim }g\left( x\right) =M}}\]

Tada vrijede pravila:

\(\color{green}{Limes\:zbroja\:i\:razlike}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }\left[ f\left( x\right) \pm g\left( x\right)\right] =L\pm M=\underset{x\rightarrow a}{\lim }f\left( x\right) \pm \underset{x\rightarrow a}{\lim }g\left( x\right) \tag{LP1}}}\]

\(\color{green}{Limes\:umnoška}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }\left[ f\left( x\right) \cdot g\left(x\right) \right] =L\cdot M=\left( \underset{x\rightarrow a}{\lim }f\left(x\right) \right) \cdot \left( \underset{x\rightarrow a}{\lim }g\left(x\right) \right) \tag{LP2}}}\]

\(\color{green}{Limes\:i\:konstanta-Posljedica \:LP2}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }cf\left( x\right) =c\underset{x\rightarrow a} {\lim }f\left( x\right) \tag{LP3}}}\]

\(\color{green}{Limes\:kvocijenta}\)

\[\color{red}{\large{\underset{x\rightarrow a}{\lim }\dfrac{f\left( x\right) }{g\left( x\right) }=\dfrac{L}{M}=\dfrac{\underset{x\rightarrow a}{\lim }f\left( x\right) }{\underset{x\rightarrow a}{\lim }g\left( x\right) };~\ \ \ \ M=\underset{x\rightarrow a}{\lim}g\left( x\right) \neq 0 \tag{LP4}}}\]