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Indeterminate Forms

An indeterminate form is a form that is encountered whose evaluation is uncertain and further analysis is required. The seven indeterminate forms which are routinely studied in calculus are as follows.

\(\displaystyle \frac{0}{0},\, \frac{\infty}{\infty},\, 0\cdot\infty,\, \infty-\infty,\, 0^0,\, 1^{\infty},\, \infty^0 \)

To determine the evaluation of an indeterminate form we may opt to algebraically “simplify” the function into another form or, in special circumstances, L’Hôpital’s Rule may be applied. 

  1. Algebraic manipulate the limit
    1. Factor the numerator and the denominator of a rational function to divide out a common factor.
      • Note that if the limit has the form \(\textstyle\frac{0}{0}\) if \(x\to a\) then the numerator and the denominator must share the common factor \((x-a)\), otherwise, the denominator and the numerator would not equal \(0\).
    2. When we are confronted with the indeterminate form \(\textstyle\infty-\infty\) we opt to do one of the following procedures.
      • Combine fractions into a single fraction and then simplify the results.
      • Combine logarithms by using the property \(\,\texttt{ln}(A)-\texttt{ln}(B)=\texttt{ln}\left(\frac{A}{B}\right)\).
    3. When square roots are present you may wish to consider using the conjugate and applying the difference of squares identity.
      • The conjugate of \(\,a+\sqrt{b}\,\) is \(\,a-\sqrt{b}\).
      • The difference of square identity is \(\,a^2-b^2=(a-b)(a+b)\).
    4. If \(x\) is growing without bounds in a rational function, consider multiplying the rational function by \(\textstyle\frac{\frac{1}{x^n}}{\frac{1}{x^n}}\), where \(n\) is the highest degree (for the polynomial in either the numerator or the denomiator) found in the rational function. Simplfiy the product then apply the limit \(\textstyle\underset{x\to\infty}{\lim}\frac{1}{x^n}=0\) to each term and simplify the limit. 
    5. Consider how the Squeeze Theorem can be used to assess the limit.
  2. Apply L’Hôpital’s Rule
    1. Only applicable if the limit corresponds to either \(\textstyle\frac{\infty}{\infty}\) or \(\textstyle\frac{0}{0}\)
    2. \(\underset{x\to a}{\lim}\frac{f(x)}{g(x)}\stackrel{\small{L’H}}{=}\underset{x\to a}{\lim}\frac{f'(x)}{g'(x)}\)
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Indeterminate Forms

Limits

A limit has an indeterminate form if the limit corresponds to one of the following two forms

  • \(\frac{0}{0}\)
  • \(\frac{\infty}{\infty}\)
  • \(0\cdot\infty\)
  • \(\infty-\infty\)
  • \(0^0\)
  • \(1^{\infty}\)
  • \(\infty^0\)
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\(\displaystyle\boldsymbol{\frac{0}{0}}\)

Limits

  • Factor the numerator and the denominator to divide out the zero terms
  • Consider how the conjugate may lead to dividing out terms
  • Apply L’Hôpital’s Rule.
  • Apply the Squeeze Theorem
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\(\displaystyle\boldsymbol{\frac{\infty}{\infty}}\)

Limits

  • Factor the numerator and the denominator to divides out the zero terms
  • Consider how the conjugate may lead to dividing out terms
  • Apply L’Hôpital’s Rule.
  • Apply the Squeeze Theorem
 Note:
  • If \(x\rightarrow\infty\) then \(x=\sqrt{x^2}\)
  • If \(x\rightarrow-\infty\) then \(x=-\sqrt{x^2}\)
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\(\displaystyle\boldsymbol{0\cdot\infty}\)

Limits

Consider how to algebraically express the function so that results leads to either \(\textstyle\frac{0}{0}\) or \(\textstyle\frac{\infty}{\infty}\)

A useful 'trick' involves unsimplifing fractions \(\displaystyle f(x)g(x)=\frac{g(x)}{\textstyle\frac{1}{f(x)}}\)
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\(\displaystyle\boldsymbol{\infty-\infty}\)

Limits

Consider how to algebraically express the function so that results leads to either \(\textstyle\frac{0}{0}\) or \(\textstyle\frac{\infty}{\infty}\)

  • If fractions are present consider combing them and simplifing the results
  • If radicals are present investigate how the conjugate can be used.
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\(\displaystyle\boldsymbol{0^0}\)

Limits

Consider how to algebraically express the function so that results leads to either \(\textstyle\frac{0}{0}\) or \(\textstyle\frac{\infty}{\infty}\)

These types of limits typically involve using logarithmic properties, in particular, \(\texttt{ln}(u^n)=n\texttt{ln}(u).\)

Applying this property often leads the indeterminate form \(0\cdot\infty\)
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\(\displaystyle\boldsymbol{1^{\infty}}\)

Limits

Consider how to algebraically express the function so that results leads to either \(\textstyle\frac{0}{0}\) or \(\textstyle\frac{\infty}{\infty}\)

These types of limits typically involve using logarithmic properties, in particular, \(\texttt{ln}(u^n)=n\texttt{ln}(u).\)

Applying this property often leads the indeterminate form \(0\cdot\infty\)
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\(\displaystyle\boldsymbol{\infty^0}\)

Limits

Consider how to algebraically express the function so that results leads to either \(\textstyle\frac{0}{0}\) or \(\textstyle\frac{\infty}{\infty}\)

These types of limits typically involve using logarithmic properties, in particular, \(\texttt{ln}(u^n)=n\texttt{ln}(u)\).

Applying this property often leads the indeterminate form \(0\cdot\infty\)

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