7. Computing Limits

b. When Limit Laws Don't Apply

Limits without 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 several tricks that will help with this process. Click on each link to to see examples of the tricks.

1. Limit Tricks

1. Factor and Cancel
2. Expand and Cancel
3. Divide by the Largest Term in the Denominator
4. Put Terms over a Common Denominator
5. Multiply by the Conjugate

As you study these tricks, you will notice that they cover all the indeterminate forms except for the exponential forms: \[ 0^0, \qquad \qquad 1^\infty, \qquad \qquad \infty^0 \] We will return to study these cases when we cover l'Hospital's Rule as an application of differentiation.