18. Sequences

b5a. Deriving Indeterminate Forms

Limits which can be done with the Limit Laws are boring! The real question is "What do we do when the Limit Laws do not apply? The situations when the Limit Laws do not apply are called indeterminate forms. We start by identifying the situations in which the Limit Laws do not apply:

• First, the Limit Laws start with the assumption that there are sequences \(a_n\) and \(b_n\) for which \(\lim\limits_{n\to\infty}a_n=L\) and \(\lim\limits_{n\to\infty}b_n=M\) where \(L\) and \(M\) are finite. So we need to look at what happens in each law when the limits \(L\) and/or \(M\) are infinite.


• Second, the Quotient Law assumes \(M=\lim\limits_{n\to\infty}b_n\ne0\). So we need to look at the situation when the the denominator approaches \(0\). If the sequence approaches \(0\) from the positive side, we write \(0^+\) for the limit in intermediate computations and if it approaches \(0\) from the negative side, we write \(0^-\).


• Third, the Power Law assumes \(M^L\) exists. So we need to look at the situations where \(M^L\) does not exist. This means the limits \(0^0\) or \(1^\infty\) or \(\infty^0\) whose values are ambiguous (See below).

With these points in mind, we look at each of the Limit Laws and identify the indeterminate forms:

• In the Addition (or Subtraction) Law, we have \[ \infty+\infty=\infty \qquad \text{and} \qquad -\infty-\infty=-\infty \] However, the expressions \[ \infty-\infty \qquad \text{and} \qquad -\infty+\infty \] can have absolutely any limit based on which term grows to \(\infty\) faster. This is the first indeterminate form which we denote by \(\infty-\infty\).


• In the Quotient Law, we need to look at the situations when \(M\) is \(\infty\), \(-\infty\) or \(0\). We will write \(M=0^+\) to mean the sequence \(b_n\) approaches \(0\) from the positive side, and \(M=0^-\) to mean the sequence \(b_n\) approaches \(0\) from the negative side. We have:
  If \(L\) is positive and finite, i.e. \(0 < L < \infty\), then \[ \dfrac{L}{\pm\infty}=0 \qquad \dfrac{L}{0^+}=\infty \qquad \text{and} \qquad \dfrac{L}{0^-}=-\infty. \] If \(L\) is negative and finite, i.e. \(-\infty < L < 0\), then \[ \dfrac{L}{\pm\infty}=0 \qquad \dfrac{L}{0^+}=-\infty \qquad \text{and} \qquad \dfrac{L}{0^-}=\infty. \] If \(L= 0\), then \[ \dfrac{0}{\pm\infty}=0. \] If \(L=\pm\infty\), then \[ \dfrac{\infty}{0^+}=\infty \qquad \dfrac{\infty}{0^-}=-\infty \qquad \dfrac{-\infty}{0^+}=-\infty \qquad \text{and} \qquad \dfrac{-\infty}{0^-}=\infty. \] That leaves the cases \(\dfrac{0^\pm}{0^\pm}\) which we call the indeterminate form \(\dfrac{0}{0}\) and the cases \(\dfrac{\pm\infty}{\pm\infty}\) which we call the indeterminate form \(\dfrac{\infty}{\infty}\).


• The Product Rule is the same as the Quotient Rule except that \(\dfrac{1}{M}\) is replaced by \(M\). So both of the indeterminate forms \(\dfrac{0}{0}\) and \(\dfrac{\infty}{\infty}\) become the indeterminate form \(0\cdot\infty\).


• Finally, the Power Law leads to three indeterminate forms:

  
  

  \(0^0\):   Since

  \[\begin{aligned} &&0^p=0 \qquad &&\text{for} \quad p>0 \\ &&0^p=\infty \qquad &&\text{for} \quad p<0 \\ \text{but} \qquad \qquad && \\ &&b^0=1 \qquad &&\text{for} \quad b>0 \end{aligned}\] the value of \(0^0\) is ambiguous and depends on how fast the base and power approach \(0\).

  
  

  \(1^\infty\):   Since

  \[\begin{aligned} &&b^\infty=\infty \qquad &&\text{for} \quad b>1 \\ &&b^\infty=0 \qquad &&\text{for} \quad b<1 \\ \text{but} \qquad \qquad && \\ &&1^p=1 \qquad &&\text{for} \quad p\ne\pm\infty \end{aligned}\] the value of \(1^\infty\) is ambiguous and depends on how fast the base approaches \(1\) and the power approaches \(\infty\).

  
  

  \(\infty^0\):   Since

  \[\begin{aligned} &&\infty^p=\infty \qquad &&\text{for} \quad p>0 \\ &&\infty^p=0 \qquad &&\text{for} \quad p<0 \\ \text{but} \qquad \qquad && \\ &&b^0=1 \qquad &&\text{for} \quad b\ne\infty \end{aligned}\] the value of \(\infty^0\) is ambiguous and depends on how fast the base approaches \(\infty\) and the power approaches \(0\).