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

$\lim_{x→x_0}f(g(x))=\lim_{u→u_0}f(u)$