The limit is the foundational concept that underlies all of calculus. Informally, the limit of \(f(x)\) as \(x\) approaches \(c\) is the value \(L\) that \(f(x)\) approaches as \(x\) gets arbitrarily close to \(c\). More precisely, we write \(\lim_{x \to c} f(x) = L\) to mean that \(f(x)\) can be made as close to \(L\) as we like by choosing \(x\) sufficiently near \(c\). This intuitive idea is captured rigorously by the epsilon-delta definition: for every \(\varepsilon > 0\), there must exist a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \varepsilon\). The rigor of this formulation is what allows calculus to be placed on solid logical footing.
A function is continuous at a point \(c\) precisely when three conditions hold: \(f(c)\) is defined, \(\lim_{x \to c} f(x)\) exists, and that limit equals \(f(c)\). Graphically, a continuous function is one whose graph can be drawn through \(c\) without lifting the pen. Discontinuities arise when one or more of these conditions fails; a particularly interesting case is the removable discontinuity, which occurs when the limit exists at \(c\) but either \(f(c)\) is undefined or \(f(c) \neq \lim_{x \to c} f(x)\). Such a discontinuity can be removed by redefining \(f(c)\) to equal the limit value. Closely related are one-sided limits, written \(\lim_{x \to c^+} f(x)\) for the right-hand limit and \(\lim_{x \to c^-} f(x)\) for the left-hand limit; the two-sided limit exists only when both one-sided limits agree.
Several important theorems build on the notion of limits. The Squeeze Theorem states that if \(g(x) \leq f(x) \leq h(x)\) near \(c\) and both \(g\) and \(h\) share the same limit \(L\) at \(c\), then \(f\) must also have limit \(L\). This is how one proves the fundamental limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), using geometric comparisons with the unit circle. A practical matter is the interpretation of infinite limits: writing \(\lim_{x \to c} f(x) = \infty\) does not mean the limit equals infinity in the usual sense; rather, \(f(x)\) grows without bound as \(x\) approaches \(c\), exceeding every positive number eventually.
When limits are taken as \(x\) grows arbitrarily large, we examine end behavior. For rational functions \(P(x)/Q(x)\) as \(x \to \infty\), the standard technique is to divide every term by the highest power of \(x\) appearing in the denominator. After this division, every term containing \(x\) in the denominator approaches 0, leaving only the ratio of leading coefficients when the numerator and denominator have the same degree. The Intermediate Value Theorem rounds out this chapter: if \(f\) is continuous on a closed interval \([a, b]\), then \(f\) attains every value between \(f(a)\) and \(f(b)\) somewhere in \((a, b)\). This theorem is foundational for many existence arguments in calculus and beyond.