lhospitals_rule

L’Hôpital’s Rule

Definition

L’Hôpital’s Rule. If $f(a)=g(a)=0$ and $g^{\prime }(a)\ne 0$, then ${\lim }_{x\to a}\frac{f(x)}{g(x)}=\frac{f^{\prime }(a)}{g^{\prime }(a)}.$

Intuition

Near $x=a$, both $f$ and $g$ are approximated by their linear terms. When both vanish at $a$, the ratio reduces to the ratio of slopes — i.e. the ratio of first derivatives evaluated at $a$.

Formal Description

L’Hôpital’s rule handles limits that produce the indeterminate forms $\frac{0}{0}$ or $\frac{\infty }{\infty }$ under direct substitution.

Derivation via Taylor series. Suppose $f$ and $g$ have Taylor series around $x=a$:

$$f(x)=f(a)+(x-a)f^{\prime }(a)+\frac{(x-a{)}^2}{2!}{f}^{\prime \prime }(a)+\dots$$ $$g(x)=g(a)+(x-a)g^{\prime }(a)+\frac{(x-a{)}^2}{2!}{g}^{\prime \prime }(a)+\dots$$

When $f(a)=g(a)=0$, both numerator and denominator contain a factor of $(x-a)$:

$$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{(x-a)\left(f^{\prime }(a)+\frac{x-a}{2!}{f}^{\prime \prime }(a)+\dots \right)}{(x-a)\left(g^{\prime }(a)+\frac{x-a}{2!}{g}^{\prime \prime }(a)+\dots \right)}=\frac{f^{\prime }(a)}{g^{\prime }(a)}.$$

Key Results

• Repeated application: if $f^{\prime }(a)=g^{\prime }(a)=0$, apply the rule again to $f^{\prime }/g^{\prime }$.
• $\frac{\infty }{\infty }$ form: the rule applies equally.
• Other indeterminate forms ($0\cdot \infty$, $\infty -\infty$, $0^0$, etc.) can often be rewritten algebraically as $\frac{0}{0}$ or $\frac{\infty }{\infty }$ before applying the rule.

Example.

$$\underset{x\to 0}{\lim }\frac{\sin (ax)}{x}\stackrel{0/0}{\to }\underset{x\to 0}{\lim }\frac{a\cos (ax)}{1}=a.$$

Applications

L’Hôpital’s rule is used to evaluate limits that arise in computing Taylor series coefficients, verifying series convergence, and simplifying indeterminate expressions in analysis.

Trade-offs

The rule requires $f$ and $g$ to be differentiable near $a$. Repeated application can cycle or fail to simplify; direct Taylor expansion is often more efficient in such cases. Forms like ${\infty }^0$ or $1^{\infty }$ require algebraic rewriting (e.g. via logarithms) before the rule can be applied.

Links

• Taylor Series
• Sequences and Series