The tangent line problem

For a circle, "the tangent at $P$" has a clean, purely geometric meaning: it is the line through $P$ perpendicular to the radius at $P$ (Fig. 10.1). For a general curve, that definition breaks down entirely — there is no "radius" to be perpendicular to. Nor does "touches but does not cross" work in general: it holds for some curves (Fig. 10.2) but fails for others where a genuine tangent line still crosses the curve at the point of tangency (Fig. 10.3); and "touches at exactly one point" works for a circle but fails for most other curves (Fig. 10.4), which a tangent line may meet again elsewhere.

The way out is to stop trying to define the tangent line directly and instead define it as a limit of secant lines. Let $P(x_0, f(x_0))$ be the point of tangency and let $Q(x_0+\Delta x,\ f(x_0+\Delta x))$ be a second point on the curve $y=f(x)$. The line through $P$ and $Q$ is a secant line, and (from the two-point slope formula) its slope is the difference quotient

where $\Delta x$ is the increment in $x$ and $\Delta y = f(x_0+\Delta x)-f(x_0)$ is the corresponding change in $y$. The beauty of this construction is that choosing $Q$ closer and closer to $P$ (i.e. letting $\Delta x \to 0$) makes $m_{\text{sec}}$ a better and better approximation to the true slope of the tangent at $P$ — turning an unsolved geometric question into a limit computation.

Worked illustration (the book's own numbers). Take $f(x)=x^2$ and $P=(1,1)$. With $\Delta x = 0.1$: $f(1.1) = 1.21$, so $\Delta y = 1.21-1=0.21$ and $m_{\text{sec}} = 0.21/0.1 = 2.1$. Tabulating $\Delta y/\Delta x$ for $\Delta x = \pm0.1, \pm0.01, \pm0.001$ gives $2.1, 2.01, 2.001$ from the right and $1.9, 1.99, 1.999$ from the left — both sequences squeezing toward $2$. Formally, $\lim_{\Delta x\to 0^-}\Delta y/\Delta x = 2 = \lim_{\Delta x\to0^+}\Delta y/\Delta x$, so $\lim_{\Delta x\to0}\Delta y/\Delta x = 2$: the tangent to $y=x^2$ at $(1,1)$ has slope $m_{\tan}=2$.

This motivates:

Definition 10.1 (Tangent line with slope $m$). Let $f$ be defined on an open interval containing $x_0$. If

$m_{\tan} = \lim_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x} = \lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}$

exists, then the line through $(x_0,f(x_0))$ with slope $m_{\tan}$ is the tangent line to the graph of $f$ at $(x_0,f(x_0))$; this same number is also called the slope of the curve at that point.

Because a point together with a slope determines exactly one line, the definition guarantees that if a tangent line exists at a point, it is unique. …