Step By Step Calculus » 9.5 - Limits of Indeterminate Forms

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When evaluating limits, we sometimes encounter forms that cannot be readily resolved using the Limit Laws, the Squeeze Theorems and the Monotone Theorems. These are known as indeterminate limits or indeterminate forms. The indeterminate forms are:
  • \frac{0}{0}\frac{0}{0}
  • \frac{\infty}{\infty}\frac{\infty}{\infty}
  • 0\cdot \infty0\cdot \infty
  • \infty-\infty\infty-\infty
  • 0^00^0
  • \infty^0\infty^0
  • 1^\infty1^\infty
There are several effective strategies for dealing with these forms, especially when dealing with rational and algebraic functions.
  • By factoring the numerator and denominator of a rational function you may be able to cancel out common factors, producing a determinate form. Long division, or even Euclidean division may be helpful in this process.
  • Multiplying by an appropriate function or ratio of a function to itself ( = 1= 1, so does not change things) can often help. This is especially useful if square roots are involved.
  • Sometimes a determinate form will not be readily present, so careful consideration of the problem may be necessary, and application of Squeeze Theorems or other theorems may help.
In order to determine limits of indeterminate forms involving trigonometric functions the fundamental limit \displaystyle \lim_{\theta\to 0}\frac{\sin\theta}{\theta}\displaystyle \lim_{\theta\to 0}\frac{\sin\theta}{\theta} will often play an important role. Using the geometry of the unit circle, and the Squeeze Convergence Theorem, we will establish that \displaystyle \lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1\displaystyle \lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1.
Later in our course we will discuss L’Hospital’s rule and Taylor polynomials, which are very powerful tools for evaluating limits of indeterminate forms.