Step By Step Calculus » 9.3 - Limit Laws

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Synopsis
Since it can be very tedious to establish convergence or divergence using the definition of limit and many problems are not amenable to the monotone and squeeze theorems, we are going to develop a set of limit laws. When we state a law for convergence/divergence as x \rightarrow ax \rightarrow a, that law can also be stated for approach to aa from either side. The equivalent statements for approach from the right or left will be obvious. In addition, all of the laws apply when the real number aa is replaced by the symbols + \infty + \infty and - \infty- \infty.
If \displaystyle\lim_{x\rightarrow a}f(x)=L_1\displaystyle\lim_{x\rightarrow a}f(x)=L_1, \displaystyle\lim_{x\rightarrow a }g(x)=L_2\displaystyle\lim_{x\rightarrow a }g(x)=L_2 and c, c_1, c_2c, c_1, c_2 are constants, then the following are true:
Convergent Linearity Law: (a) \displaystyle\lim_{x\rightarrow a}(cf(x)) = cL_1\displaystyle\lim_{x\rightarrow a}(cf(x)) = cL_1, (b) \displaystyle\lim_{x\rightarrow a}(f(x)+g(x)) = L_1+L_2\displaystyle\lim_{x\rightarrow a}(f(x)+g(x)) = L_1+L_2. Combining (a) and (b), we have (c) \displaystyle\lim_{x\rightarrow a}(c_1f(x)+c_2g(x)) = c_1L_1+c_2L_2\displaystyle\lim_{x\rightarrow a}(c_1f(x)+c_2g(x)) = c_1L_1+c_2L_2.
Convergent Quotient Law: \displaystyle\lim_{x\rightarrow a} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}\displaystyle\lim_{x\rightarrow a} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}, provided that L_2\neq 0L_2\neq 0.
Convergent Product Law: \displaystyle\lim_{x\rightarrow a}f(x)g(x) = L_1\cdot L_2\displaystyle\lim_{x\rightarrow a}f(x)g(x) = L_1\cdot L_2. [If \displaystyle\lim_{x\rightarrow a }f(x)\displaystyle\lim_{x\rightarrow a }f(x) is indeterminate but bounded and L_2=0L_2=0, then \displaystyle\lim_{x\rightarrow a}f(x)g(x) = 0.\displaystyle\lim_{x\rightarrow a}f(x)g(x) = 0.]
Convergent Power Law: \displaystyle\lim_{x\rightarrow a}\left( f(x)\right)^{p} = L_1^{p}\displaystyle\lim_{x\rightarrow a}\left( f(x)\right)^{p} = L_1^{p} for p\in\mathbb{R}p\in\mathbb{R} if \left( f(x)\right) ^{p}\left( f(x)\right) ^{p} and L_1^pL_1^p make sense. They make sense for example when f(x)>0f(x)>0 and L_1>0L_1>0.
\displaystyle \left\{\begin{array}{ll} \displaystyle \lim_{t\rightarrow b^{+}} f(h(t))=L_1=\lim_{x\to a^+}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{+}} h(t)=a \textrm{ and $h(t)$ is monotonically increasing}\\ \displaystyle\lim_{t\rightarrow b^{+}} f(h(t))=L_1=\lim_{x\to a^-}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{+}} h(t)=a \textrm{ and $h(t)$ is monotonically decreasing}\\ \displaystyle\lim_{t\rightarrow b^{-}} f(h(t))=L_1=\lim_{x\to a^-}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{-}} h(t)=a \textrm{ and $h(t)$ is monotonically increasing}\\ \displaystyle\lim_{t\rightarrow b^{-}} f(h(t))=L_1=\lim_{x\to a^+}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{-}} h(t)=a \textrm{ and $h(t)$ is monotonically decreasing} \end{array}\right\}.
\displaystyle \left\{\begin{array}{ll} \displaystyle \lim_{t\rightarrow b^{+}} f(h(t))=L_1=\lim_{x\to a^+}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{+}} h(t)=a \textrm{ and $h(t)$ is monotonically increasing}\\ \displaystyle\lim_{t\rightarrow b^{+}} f(h(t))=L_1=\lim_{x\to a^-}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{+}} h(t)=a \textrm{ and $h(t)$ is monotonically decreasing}\\ \displaystyle\lim_{t\rightarrow b^{-}} f(h(t))=L_1=\lim_{x\to a^-}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{-}} h(t)=a \textrm{ and $h(t)$ is monotonically increasing}\\ \displaystyle\lim_{t\rightarrow b^{-}} f(h(t))=L_1=\lim_{x\to a^+}f(x) & \textrm{ if } \displaystyle\lim_{t\to b^{-}} h(t)=a \textrm{ and $h(t)$ is monotonically decreasing} \end{array}\right\}.
Convergent Function-to-Sequence Law: \displaystyle\lim_{n\rightarrow \infty} s_n=L_1\displaystyle\lim_{n\rightarrow \infty} s_n=L_1 if s_n=f(n)s_n=f(n) and a=\infty.a=\infty.
There are three other important convergent limit laws that we often use. These are:
Convergent Sequence-to-Function Law: \displaystyle \lim_{x\to a^+}f(x)=L\;\;\left(\lim_{x\to a^-}f(x)=L\right)\displaystyle \lim_{x\to a^+}f(x)=L\;\;\left(\lim_{x\to a^-}f(x)=L\right) if and only if \displaystyle\lim_{n\to\infty}f(x_n)=L\displaystyle\lim_{n\to\infty}f(x_n)=L for all sequences \left\{x_n\right\}_{n=1}^{\infty}\left\{x_n\right\}_{n=1}^{\infty} such that \displaystyle\lim_{n\to\infty}x_n=a\displaystyle\lim_{n\to\infty}x_n=a and x_n>a\;\forall nx_n>a\;\forall n (x_n<a\;\;\forall nx_n<a\;\;\forall n).
Convergent Even Law: \displaystyle \lim_{x\to -a^{\mp}} f(x)=\lim_{x\to a^{\pm}} f(x)\displaystyle \lim_{x\to -a^{\mp}} f(x)=\lim_{x\to a^{\pm}} f(x) when ff is even.
Convergent Odd Law: \displaystyle \lim_{x\to -a^{\mp}} f(x)=-\lim_{x\to a^{\pm}} f(x)\displaystyle \lim_{x\to -a^{\mp}} f(x)=-\lim_{x\to a^{\pm}} f(x) when ff is odd.
By \displaystyle \lim_{x\to-a^{\mp}}f(x)=-\lim_{x\to a^{\pm}} f(x)\displaystyle \lim_{x\to-a^{\mp}}f(x)=-\lim_{x\to a^{\pm}} f(x), we mean both \displaystyle \lim_{x\to -a^-}f(x)=-\lim_{x\to a^+}f(x)\displaystyle \lim_{x\to -a^-}f(x)=-\lim_{x\to a^+}f(x) and \displaystyle \lim_{x\to -a^+}f(x)=-\lim_{x\to a^-}f(x)\displaystyle \lim_{x\to -a^+}f(x)=-\lim_{x\to a^-}f(x).
We can also state and prove divergence laws. Suppose that \displaystyle\lim_{x\rightarrow a}f(x)= +\infty \displaystyle\lim_{x\rightarrow a}f(x)= +\infty , \displaystyle\lim_{x\rightarrow a }g(x)=L_2\displaystyle\lim_{x\rightarrow a }g(x)=L_2 (L_2L_2 can be +\infty+\infty and aa can be \pm\infty\pm\infty) and let cc be any real constant.
Divergent Linearity Law:
\displaystyle \displaystyle\lim_{x\rightarrow a}(cf(x)+g(x))=\left\{ \begin{array}{ll} \infty & \textrm{if }c>0 \\ L_2 & \textrm{if }c=0\\ \left\{\begin{array}{ll} -\infty & \textrm{if } L_2\neq \infty \\ \textrm{indeterminate} & \textrm{if } L_2=\infty \end{array}\right\}& \textrm{if }c<0 \end{array} \right\}
\displaystyle \displaystyle\lim_{x\rightarrow a}(cf(x)+g(x))=\left\{ \begin{array}{ll} \infty & \textrm{if }c>0 \\ L_2 & \textrm{if }c=0\\ \left\{\begin{array}{ll} -\infty & \textrm{if } L_2\neq \infty \\ \textrm{indeterminate} & \textrm{if } L_2=\infty \end{array}\right\}& \textrm{if }c<0 \end{array} \right\}
Divergent Quotient Law:
\displaystyle \displaystyle\lim_{x\rightarrow a^{\pm }}\frac{f(x)}{g(x)}=\left\{ \begin{array}{ll} \infty & \text{if }L_2\in(0,\infty) \\ &\\ \left\{ \begin{array}{ll} \infty & \text{if } g(x)\geq 0 \text{ near }a \textrm{ on appropriate side}\\ -\infty & \text{if } g(x)< 0 \text{ near }a \textrm{ on appropriate side} \end{array} \right\} & \text{if } L_2=0 \\ & \\ -\infty & \text{if }L_2\in(-\infty,0) \\ \textrm{indeterminate} &\textrm{if } L_2=\infty \end{array} \right\}
\displaystyle \displaystyle\lim_{x\rightarrow a^{\pm }}\frac{f(x)}{g(x)}=\left\{ \begin{array}{ll} \infty & \text{if }L_2\in(0,\infty) \\ &\\ \left\{ \begin{array}{ll} \infty & \text{if } g(x)\geq 0 \text{ near }a \textrm{ on appropriate side}\\ -\infty & \text{if } g(x)< 0 \text{ near }a \textrm{ on appropriate side} \end{array} \right\} & \text{if } L_2=0 \\ & \\ -\infty & \text{if }L_2\in(-\infty,0) \\ \textrm{indeterminate} &\textrm{if } L_2=\infty \end{array} \right\}
\displaystyle \displaystyle and \displaystyle \displaystyle\lim_{x\rightarrow a^{\pm }}\frac{g(x)}{f(x)}=\left\{\begin{array}{ll}0 & \textrm{if } L_2\notin \{\infty, -\infty\}\\ \textrm{indeterminate} & \textrm{if } L_2\in\{\infty, -\infty\}\end{array}\right\}\displaystyle \displaystyle\lim_{x\rightarrow a^{\pm }}\frac{g(x)}{f(x)}=\left\{\begin{array}{ll}0 & \textrm{if } L_2\notin \{\infty, -\infty\}\\ \textrm{indeterminate} & \textrm{if } L_2\in\{\infty, -\infty\}\end{array}\right\}
Divergent Product Law: \displaystyle\lim_{x\rightarrow a^{\pm }}f(x)\cdot g(x)=\left\{ \begin{array}{ll} \infty & \text{if }L_2>0 \\ \textrm{indeterminate} &\textrm{if } L_2=0\\ -\infty & \text{if }L_2<0 \end{array} \right\} \displaystyle\lim_{x\rightarrow a^{\pm }}f(x)\cdot g(x)=\left\{ \begin{array}{ll} \infty & \text{if }L_2>0 \\ \textrm{indeterminate} &\textrm{if } L_2=0\\ -\infty & \text{if }L_2<0 \end{array} \right\}
Divergent Power Law: \displaystyle\lim_{x\rightarrow a^{\pm }}\left( f(x)\right) ^{p}=\infty \displaystyle\lim_{x\rightarrow a^{\pm }}\left( f(x)\right) ^{p}=\infty if p>0p>0 .
The Divergent Quotient Law also includes the case when L_1\in(-\infty,\infty)L_1\in(-\infty,\infty) and L_2 = 0L_2 = 0:
\displaystyle \lim_{x\rightarrow a^{\pm }}\frac{f(x)}{g(x)}=\left\{ \begin{array}{ll} -\infty & \textrm{ if } L_1<0, L_1\neq -\infty, g(x)\geq 0 \textrm{ near } a \textrm{ on appropriate side}\\ \infty & \textrm{ if } L_1<0, L_1\neq -\infty, g(x)< 0 \textrm{ near } a \textrm{ on appropriate side}\\ \infty & \textrm{ if } L_1\geq0, L_1\neq \infty, g(x)\geq 0 \textrm{ near } a \textrm{ on appropriate side}\\ -\infty & \textrm{ if } L_1\geq0, L_1\neq \infty, g(x)< 0 \textrm{ near } a \textrm{ on appropriate side}\\ \end{array}\right\}
\displaystyle \lim_{x\rightarrow a^{\pm }}\frac{f(x)}{g(x)}=\left\{ \begin{array}{ll} -\infty & \textrm{ if } L_1<0, L_1\neq -\infty, g(x)\geq 0 \textrm{ near } a \textrm{ on appropriate side}\\ \infty & \textrm{ if } L_1<0, L_1\neq -\infty, g(x)< 0 \textrm{ near } a \textrm{ on appropriate side}\\ \infty & \textrm{ if } L_1\geq0, L_1\neq \infty, g(x)\geq 0 \textrm{ near } a \textrm{ on appropriate side}\\ -\infty & \textrm{ if } L_1\geq0, L_1\neq \infty, g(x)< 0 \textrm{ near } a \textrm{ on appropriate side}\\ \end{array}\right\}
When a = + \infty \; a = + \infty \; or \; - \infty \; - \infty, it obviously does not make sense to write x\to\infty^{\pm}x\to\infty^{\pm}. In this case we consider only x\to +\infty^-x\to +\infty^- (i.e., xx approaches +\infty+\infty from the left). Similarly, when x\to -\inftyx\to -\infty, we will consider only x\to -\infty^+x\to -\infty^+ (i.e., xx approaches -\infty-\infty from the right).