3.5 · Linear approximation and L’Hôpital’s Rule

Use the tangent as a local model

Near $\displaystyle x=a$, replace $\displaystyle f(x)$ by its tangent-line model $\displaystyle L(x)=f(a)+f'(a)(x-a)$. Choose a nearby input where values and derivatives are easy to evaluate.

For $\displaystyle f(x)=\sqrt{x}$ near $\displaystyle a=4$, $\displaystyle L(x)=2+(x-4)/4$. The estimate at $\displaystyle 4.1$ is $\displaystyle 2.025$. This is an approximation, so use an approximation symbol when reporting the function value.

$$L(x)=f(a)+f'(a)(x-a)$$

Use concavity to describe the error

Where a graph is concave down throughout the interval between $\displaystyle a$ and the target, the tangent lies above the curve, giving an overestimate. Concave up gives an underestimate.

The differential $\displaystyle dy=f'(a)\,dx$ estimates the actual change $\displaystyle f(a+dx)-f(a)$; they are not generally identical. Estimates tend to improve as the target approaches the expansion input.

Check the form before differentiating a quotient

L’Hôpital’s Rule applies to eligible $\displaystyle 0/0$ or $\displaystyle \infty/\infty$ quotients under differentiability and derivative-limit conditions. For $\displaystyle \lim_{x\to0}(e^x-1)/x$, both numerator and denominator approach zero, and the derivative quotient tends to $\displaystyle 1$.

The rule uses the quotient of the separate derivatives, not the quotient rule. For a product or difference, first rewrite into an eligible quotient if possible. If substitution already gives a finite ordinary quotient, do not replace it by derivatives.

In an interval near the target, the numerator and denominator must be differentiable, with the denominator derivative nonzero. The quotient of derivatives must have the required finite or infinite limit. Check these conditions before replacing the quotient; use the matching one-sided version if approaching an endpoint.

$$\lim_{x\to0}\frac{e^x-1}{x}=\lim_{x\to0}\frac{e^x}{1}=1$$