1. Why L'Hospital's Rule Is Needed
Sometimes, evaluating a limit leads to indeterminate forms like 0/0 or ∞/∞. These forms do not reveal the true value of the limit. L'Hospital's Rule gives a direct way to resolve such limits using derivatives.
2. What L'Hospital's Rule Says
L'Hospital's Rule applies only when an expression produces 0/0 or ∞/∞ after substitution. The rule lets us differentiate the numerator and denominator separately and then take the limit again.
2.1. Formula
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{if the original limit gives } \frac{0}{0} \text{ or } \frac{\infty}{\infty}
3. Conditions for Applying L'Hospital's Rule
Before applying the rule, certain conditions must hold.
3.1. Required Conditions
- The original limit must give 0/0 or ∞/∞.
- Both f and g must be differentiable near the point of interest.
- The derivative of g(x) must not be zero near that point.
4. Examples Using L'Hospital's Rule
These examples show how the rule simplifies difficult limits.
4.1. Example 1 — 0/0 Form
Evaluate:
\lim_{x \to 0} \frac{\sin x}{x}
Substitution gives 0/0.
Differentiate:
f'(x) = \cos x, \quad g'(x) = 1
\lim_{x \to 0} \frac{\cos x}{1} = 1
4.2. Example 2 — ∞/∞ Form
Evaluate:
\lim_{x \to \infty} \frac{e^x}{x}
Substitution gives ∞/∞.
Differentiate:
f'(x) = e^x, \quad g'(x) = 1
\lim_{x \to \infty} e^x = \infty
4.3. Example 3 — Cancelled Growth
Evaluate:
\lim_{x \to \infty} \frac{\ln x}{x}
Substitution gives ∞/∞.
f'(x) = \frac{1}{x}, \quad g'(x) = 1
\lim_{x \to \infty} \frac{1/x}{1} = 0
5. Repeated Use of L'Hospital's Rule
Sometimes the first application still gives an indeterminate form. In such cases, L'Hospital's Rule can be applied repeatedly until the form is resolved.
5.1. Example
\lim_{x \to 0} \frac{1 - \cos x}{x^2}
First substitution gives 0/0.
Apply the rule twice to reach the final value of 1/2.
6. Forms That Can Be Converted for L'Hospital
Even if a limit is not directly 0/0 or ∞/∞, it can sometimes be rewritten to fit those forms.
6.1. Examples of Convertible Forms
- 0 · ∞ → rewrite as a quotient
- ∞ − ∞ → combine into a single fraction
- 00, ∞0, 1∞ → apply logarithms
7. Why L'Hospital's Rule Matters
The rule provides a powerful shortcut for handling tough limits. It reduces complex expressions to simpler forms by using derivatives, helping reveal the true behaviour of functions near difficult points.