Spot an indeterminate form: the 4 classics

The 4 indeterminate forms

$\boxed{\infty - \infty} \quad \boxed{\dfrac{0}{0}} \quad \boxed{\dfrac{\infty}{\infty}} \quad \boxed{0 \times \infty}$

When you face one, you can’t conclude directly. You need to transform the expression to lift the indeterminacy.

Cases that are not indeterminate

• $\infty + \infty = \infty$
• $\infty \times \infty = \infty$
• $\dfrac{k}{0^+} = \pm\infty$ (depending on sign of $k$)
• $\dfrac{k}{\pm\infty} = 0$

How to lift an IF

• Factor out the dominant term (useful at $\pm\infty$).
• Expand or conjugate (to remove a root).
• Simplify a common factor (typical when $x \to a$).

A quick example

$\displaystyle \lim_{x \to +\infty} (x^2 - x)$ is of form $\infty - \infty$. Factor: $x^2(1 - \tfrac{1}{x}) \to +\infty \times 1 = +\infty$.

The reflex

Before a limit, always write out the form obtained before concluding. If it’s indeterminate, say so explicitly and transform.

?Your turn

Which of the following is an indeterminate form?

A.∞ + ∞B.0 × ∞C.1 / ∞D.∞ × 5

Check