Indeterminate forms

The rules do not always apply directly: certain combinations produce expressions with no definite value.

In summary — Rule 2: indeterminate forms

When applying the rules produces an expression of the type $\frac{0}{0},\quad \frac{\infty}{\infty},\quad \infty-\infty,\quad 0\cdot\infty,\quad 1^\infty,\quad \infty^0,\quad 0^0,$ the limit cannot be computed directly: the expression must be transformed. The strategies fall into four routes:

1. Substitute equivalent functions (approximations for $x\to x_0$);
2. Simplify using the hierarchy of infinities/infinitesimals;
3. Factorise the numerator and denominator and simplify;
4. Rationalise (especially with roots, in $\infty-\infty$ forms).

The following diagram summarises the choice of strategy according to the indeterminate form.

The four strategies for resolving an indeterminate form.

Links

Topics: Limits
Concepts: Indeterminate forms · Limit
Skills: Reasoning by cases