Indeterminate Forms

When direct substitution is tried on a limit and produces an expression like $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, the result is called an indeterminate form — it does not mean the limit fails to exist, and it certainly does not mean the limit equals $0$, $1$ or $\infty$ just because the symbols suggest it. It means direct substitution has simply failed to answer the question, and a different technique must be used to find the true value (which may turn out to be any finite number, or may genuinely not exist).

The $\dfrac{0}{0}$ form typically arises in a ratio of polynomials (or expressions containing a surd) where both numerator and denominator vanish at the same point — resolved by factorisation or rationalisation, exactly as described in the previous section.

The $\dfrac{\infty}{\infty}$ form arises when both numerator and denominator grow without bound as $x \to \infty$ (or $x \to -\infty$) — typically a ratio of two polynomials. The standard technique is to divide every term of both numerator and denominator by the highest power of $x$ appearing in the expression, so that every term of the form $\dfrac{1}{x^k}$ tends to $0$ and only the leading terms survive. For a ratio of two polynomials of degree $m$ (numerator) and $n$ (denominator) as $x \to \infty$: