Tangent-line estimates and L’Hôpital’s rule

Near $x = a$, a differentiable function stays close to its tangent line. The local linear approximation is $L(x) = f(a) + f'(a)(x - a)$, and $f(x) \approx L(x)$ for $x$ near $a$. The closer $x$ is to $a$, the better the estimate usually is.

Too high or too low? If $f'' > 0$ between $a$ and $x$, the graph is concave up and lies above its tangent line, so $L(x)$ is an underestimate. If $f'' < 0$ there, the graph is concave down and $L(x)$ is an overestimate.

L’Hôpital’s rule. Suppose $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, or both limits are infinite, and $f$ and $g$ are differentiable near $a$ with $g'(x) \neq 0$ near $a$ (except possibly at $a$). Then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, as long as the second limit exists or is infinite. The rule also works for limits as $x \to \infty$.

Differentiate the top and the bottom separately. This is not the quotient rule. If the new limit is still 0/0, you may apply the rule again.

Justify an over- or underestimate by naming the sign of $f''$ on the interval, the concavity it gives, and where the tangent line sits: "Because $f'' < 0$ on the interval, the graph is concave down and lies below its tangent line, so the estimate is too high."

Justify using L’Hôpital’s rule by showing that the top and the bottom both approach 0 (or both grow without bound). Write the limits, then write the new quotient of derivatives.