Limits

The Core Idea of Limits

The limit is the foundational concept of calculus. It describes the value a function approaches as its input nears a point, ignoring what happens at that point.

$$\lim_{x \to a} f(x) = L$$

As $x$ gets arbitrarily close to $a$ (without equalling it), $f(x)$ nears $L$. This examines the neighborhood of $a$, ignoring $a$ itself.

One-sided limits approach from one direction: $x \to a^-$ (left) and $x \to a^+$ (right). The two-sided limit exists only when both one-sided limits exist and agree.

Indeterminate Forms

Limits shine on indeterminate forms like $\frac{0}{0}$. Consider $\lim_{x \to 2}\frac{x^2 - 4}{x - 2}$. Direct substitution yields $\frac{0}{0}$. Factoring and cancelling leaves $x + 2$, giving a limit of $4$. The function is undefined at $x = 2$, yet the limit exists.

Limits obey clean algebra of limits rules for sums, products, and quotients.