User:Foxjwill/Limit Notes From Wikipedia, the free encyclopedia Limit laws lim x → a [ f ( x ) + g ( x ) ] = lim x → a f ( x ) + lim x → a g ( x ) {\displaystyle \lim _{x\rightarrow a}\left[f(x)+g(x)\right]=\lim _{x\rightarrow a}f(x)+\lim _{x\rightarrow a}g(x)} lim x → a [ f ( x ) − g ( x ) ] = lim x → a f ( x ) − lim x → a g ( x ) {\displaystyle \lim _{x\rightarrow a}\left[f(x)-g(x)\right]=\lim _{x\rightarrow a}f(x)-\lim _{x\rightarrow a}g(x)} lim x → a [ c f ( x ) ] = c lim x → a f ( x ) {\displaystyle \lim _{x\rightarrow a}\left[cf(x)\right]=c\lim _{x\rightarrow a}f(x)} lim x → a [ f ( x ) g ( x ) ] = lim x → a f ( x ) ⋅ lim x → a g ( x ) {\displaystyle \lim _{x\rightarrow a}\left[f(x)g(x)\right]=\lim _{x\rightarrow a}f(x)\cdot \lim _{x\rightarrow a}g(x)} lim x → a [ f ( x ) g ( x ) ] = lim x → a f ( x ) lim x → a g ( x ) if lim x → a g ( x ) ≠ 0 {\displaystyle \lim _{x\rightarrow a}\left[{\frac {f(x)}{g(x)}}\right]={\frac {\lim _{x\rightarrow a}f(x)}{\lim _{x\rightarrow a}g(x)}}{\mbox{ if }}\lim _{x\rightarrow a}g(x)\neq 0} lim x → a [ f ( x ) ] n = [ lim x → a f ( x ) ] n {\displaystyle \lim _{x\rightarrow a}\left[f(x)\right]^{n}=\left[\lim _{x\rightarrow a}f(x)\right]^{n}} lim x → a f ( x ) n = lim x → a f ( x ) n {\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{f(x)}}={\sqrt[{n}]{\lim _{x\rightarrow a}f(x)}}} Special limits lim x → a c = c {\displaystyle \lim _{x\rightarrow a}c=c} lim x → a x = a {\displaystyle \lim _{x\rightarrow a}x=a} lim x → a x n = a n {\displaystyle \lim _{x\rightarrow a}x^{n}=a^{n}} lim x → a x n = a n {\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{x}}={\sqrt[{n}]{a}}} Limits by direct substitution If f {\displaystyle f\!} is a polynomial or a rational function and a {\displaystyle a\!} is in the domain of f {\displaystyle f\!} , then lim x → a f ( x ) = f ( a ) {\displaystyle \lim _{x\rightarrow a}f(x)=f(a)} Derivitive The derivative of a function f {\displaystyle f\!} at a number a {\displaystyle a\!} , denoted by f ′ ( a ) {\displaystyle f'(a)\!} is f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h {\displaystyle f'(a)=\lim _{h\rightarrow 0}{\frac {f(a+h)-f(a)}{h}}} if this limit exists. Related Articles
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