Limit laws
Summary
- For all laws, \(c\) is a constant and
\(\lim_{x→x_0}f(x)=L_1and\lim_{x→x_0}g(x)=L_2\)
- Uniqueness of limit. If
\(\lim_{x→x_0}f(x)\)
- exists, it is unique.
- Limit of a constant.
\(\lim_{x→x_0}c=c\)
- Limit of \(x\)
\(\lim_{x→x_0}x=x_0\)
- Addition law
\(\lim_{x→x_0}\left( f\left( x \right)+g\left( x \right) \right)=L_1+L_2\)
- Subtraction law
\(\lim_{x→x_0}\left( f\left( x \right)-g\left( x \right) \right)=L_1-L_2\)
- Constant law
\(\lim_{x→x_0}cf(x)=cL_1\)
- Multiplication law
\(\lim_{x→x_0}(f(x)⋅g(x))=L_1⋅L_2\)
- Division law). If \(L_2≠0\) ,
\(\lim_{x→x_0} \frac{f\left( x \right)}{g\left( x \right)}= \frac{L_1}{L_2}\)
- Positive power law
\(\lim_{x→x_0}{\left( f\left( x \right) \right)}^n=L_1^n\)
- Negative power law. If \(n\) is a positive integer and \(L_1≠0\) then
\(\lim_{x→x_0}{\left( f\left( x \right) \right)}^{-n}=L_1^{-n}\)
- n’th root law. If \(n\) is a positive integer and \(L_1>0\) then
\(\lim_{x→x_0}\sqrt[n]{f(x)}=\sqrt[n]{L_1}\)
- Squeeze or sandwich law 1. If \(f(x)≤g(x)≤h(x)\) for all \(x\) in an open interval \((a,b)\) that contains \(x_0\) except possibly at \(x=x_0\) and
\(\lim_{x→x_0}f\left( x \right)=L and \lim_{x→x_0}h(x)=L\)
- then
\(\lim_{x→x_0}g(x)=L\)
- Squeeze or sandwich law 2. If \(f(x)=g(x)\) for all \(x\) in an open interval \((a,b)\) that contains \(x_0\) except possibly at \(x=x_0\) then
\(\lim_{x→x_0}f(x)=\lim_{x→x_0}g(x)\)
- Composition law. \(g:A→B\) , \(f:B→C\) and \(f∘g:A→C\) is the composition, \((f∘g)(x)=f(g(x))\) . If
\(\lim_{x→x_0}g(x)=L\)
- and \(f\) is continuous at \(u=L\) then
\(\lim_{x→x_0}f(g(x))=f(L)\)
- Change of variables. \(g:A→B\) , \(f:B→C\) and \(f∘g:A→C\) is the composition, \((f∘g)(x)=f\left( g\left( x \right) \right)\) and
\(\lim_{x→x_0}g(x)=u_0\)
- If \(g\) does not take the value \(u_0\) in an open interval \((a,b)\) containing \(x_0\) except possibly at \(x_0\) or \(f\) is continuous at \(u=u_0\) then
\(\lim_{x→x_0}f(g(x))=\lim_{u→u_0}f(u)\)