7. Computing Limits

d. Limits at Infinity

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

When this happens, we need to algebraically manipulate the limit before applying the Limit Laws. Here are some old tricks and some new tricks but applied to limits at infinity. Click on each link to see examples of the tricks.

3. When Limit Laws Don't Apply

Limits without Laws - Limit Tricks

1. Divide by the Largest Term in the Denominator
2. Put Terms over a Common Denominator
3. Multiply by the Conjugate
4. Replace Variable by its Reciprocal

Again, these tricks do not cover the exponential indeterminate forms: \[ 0^0, \qquad \qquad 1^\infty \qquad \qquad \infty^0 \] which will be covered using l'Hospital's Rule as an application of differentiation.