Sometimes substituting the limiting values into an expression produces a symbolic form that does not determine the limit. Such a result is called an indeterminate form.

For example,

$$\frac00$$

is indeterminate. It does not mean that the limit is zero or undefined in every case. Different functions can produce the same form while having completely different limits.

Why $0/0$ is indeterminate

As $x\to1$,

$$\frac{x^2-1}{x-1}$$

has the form $0/0$, yet factoring gives

$$\frac{(x-1)(x+1)}{x-1}=x+1$$

for $x\ne1$, so the limit is $2$.

By contrast,

$$\frac{x-1}{(x-1)^2}=\frac1{x-1}$$

does not approach a finite value. The same symbolic form therefore contains too little information by itself.

Common indeterminate forms

Important forms include

$$\frac00,\qquad \frac{\infty}{\infty},\qquad 0\cdot\infty,\qquad \infty-\infty,$$

and the exponential forms

$$0^0,\qquad1^\infty,\qquad\infty^0.$$

These are warnings that the expression must be transformed or analyzed further.

Rewriting the problem

Factoring, rationalizing, combining fractions or rewriting products and powers can expose the actual limiting behavior. Later techniques such as asymptotic comparisons and L'Hôpital's rule provide additional tools for appropriate cases.