18. Sequences

b5. Limits Without Limit Laws

When the Limit Laws cannot be applied directly, the limit will have one of the seven indeterminate forms: \[\begin{aligned} \dfrac{0}{0}, \qquad \qquad \dfrac{\infty}{\infty}, \qquad &\qquad 0\cdot\infty, \qquad \qquad \infty-\infty, \\[5pt] 0^0, \qquad \qquad &1^\infty, \qquad \qquad \infty^0 \end{aligned}\] How do we derive these indeterminate forms?

When this happens, we need to algebraically manipulate the limit before applying the Limit Laws. There are four classes of indeterminate forms and corresponding tricks for simplifying the limit. In general, the process of finding the limit of a sequence is essentially the same as finding the limit at infinity of a continuous function. So the trick is frequently to manipulate the limit into a form where l'Hopital's Rule can be applied. Click on each link to read about the method used for each class of indeterminate form.

Indeterminate Forms

1. \(\large \dfrac{0}{0} \quad \text{and} \quad \dfrac{\infty}{\infty}\)
2. \(\large0\cdot\infty\)
3. \(\large\infty-\infty\)
4. \(\large0^0 \quad \text{and} \quad 1^\infty \quad \text{and} \quad \infty^0\)