What are indeterminate forms 0/0 and ∞/∞, and how does L'Hospital's rule evaluate them?

In one line

When a limit takes the form \(\frac00\) or \(\frac\infty\infty\), L'Hospital's rule says \(\lim\dfrac{f(x)}{g(x)}=\lim\dfrac{f'(x)}{g'(x)}\), provided the new limit exists.

Explained simply

\(\frac00\) is not a number; it is a question. The answer depends on how fast the top and bottom go to zero. If the top goes to zero twice as fast as the bottom, the ratio goes to 0; if they go at the same rate, the ratio is finite. Derivatives measure these rates, so comparing derivatives settles the race. Example: \(\dfrac{\sin x}x\to1\) because near 0, \(\sin x\) and \(x\) shrink at the same rate.

Technically, if \(f(a)=g(a)=0\) (or both tend to \(\infty\)), \(f,g\) are differentiable near \(a\) and \(g'(x)\ne0\) there, then \(\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}\). Apply repeatedly while the form stays \(\frac00\) or \(\frac\infty\infty\). This is a self-learning topic.

How it works

1. Substitute \(x=a\) to confirm the form is \(\frac00\) or \(\frac\infty\infty\).
2. Differentiate numerator and denominator separately (not by the quotient rule).
3. Substitute again. If still indeterminate, repeat.
4. Simplify using standard limits where helpful, such as \(\lim\frac{\sin x}x=1\).

Key formula

\(\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\) for forms \(\frac00\), \(\frac\infty\infty\).

Quick example

\(\displaystyle\lim_{x\to0}\frac{\sin x-x}{x^3}\) (form \(\frac00\)).

Step 1: \(\dfrac{\cos x-1}{3x^2}\), still \(\frac00\). Step 2: \(\dfrac{-\sin x}{6x}\), still \(\frac00\). Step 3: \(\dfrac{-\cos x}{6}\to-\dfrac16\).

So the limit is \(-\frac16\). This matches the series \(\sin x=x-\frac{x^3}6+\dots\).

\(\frac\infty\infty\) example: \(\displaystyle\lim_{x\to\infty}\frac{\log x}{x}=\lim\frac{1/x}{1}=0\): \(x\) grows faster than \(\log x\).

Common mistakes

• Using the quotient rule on \(\frac fg\); the rule differentiates top and bottom separately.
• Applying the rule when the form is not indeterminate, e.g. \(\frac{1}{0}\) or \(\frac{2}{3}\).
• Continuing to differentiate after the form is no longer \(\frac00\).

How MU asks it

As a self-learning topic it may appear as a 3-mark Q.1 part: "Evaluate \(\displaystyle\lim_{x\to0}\frac{e^x-1-x}{x^2}\)" (answer \(\frac12\)) or "\(\displaystyle\lim_{x\to0}\frac{x-\sin x}{\tan^3x}\)" (answer \(\frac16\)). State the form at each step, differentiate separately, and give the final value.