Limits & Continuity

1. Introduction

The limit is the single idea on which all of calculus is built. Derivatives are limits of difference quotients, definite integrals are limits of Riemann sums, and infinite series are limits of partial sums. Before we can talk meaningfully about instantaneous rates of change or accumulated area, we need a precise way to describe what it means for a function to approach a value.

Informally, the statement $\lim_{x\to a} f(x) = L$ says: as $x$ gets closer and closer to $a$ (but is never equal to $a$), the outputs $f(x)$ get arbitrarily close to the number $L$. The crucial subtlety is that the limit describes the behavior near $a$, not the value at $a$. A function can have a limit at a point where it is undefined, and it can be defined at a point where its limit disagrees with its value.

Continuity then captures the functions that behave "nicely": those with no jumps, holes, or blow-ups, where the limit and the value coincide. Continuity is the hypothesis behind the most important existence theorems in calculus — the Intermediate Value Theorem (IVT) and the Extreme Value Theorem (EVT) — and it is the property that lets us evaluate most limits by simple substitution.

By the end of this article you will be able to compute limits algebraically, recognize and resolve indeterminate forms with L'Hôpital's rule, analyze asymptotic behavior, prove simple limits with the $\epsilon$–$\delta$ definition, and rigorously test functions for continuity.

The logical flow of this chapter mirrors a standard Calculus I syllabus: we begin with the intuitive and formal definitions of limits, develop the algebraic tools (limit laws, squeeze theorem), classify discontinuities and define continuity, then apply continuity to the IVT and EVT, and finally tackle limits at infinity, asymptotes, and L'Hôpital's rule for indeterminate forms. Each named theorem plays a distinct role: the squeeze theorem handles oscillation and trig limits, the IVT guarantees roots and intermediate values, and the EVT underpins optimization on closed intervals.

2. Core Concepts

2.1 The Intuitive Limit

We write $\lim_{x\to a} f(x) = L$ when the values of $f(x)$ can be made as close to $L$ as desired by taking $x$ sufficiently close to (but not equal to) $a$. The phrase "not equal to $a$" matters: the limit ignores what happens at $a$ entirely. For example, the functions $f(x) = \frac{x^2 - 1}{x - 1}$ and $g(x) = x + 1$ agree everywhere except at $x = 1$, where $f$ is undefined, yet both have limit $2$ as $x \to 1$.

2.2 The Formal $\epsilon$–$\delta$ Definition

The precise definition removes all vagueness from "close." We say $\lim_{x\to a} f(x) = L$ if:

for every $\epsilon > 0$ there exists a $\delta > 0$ such that whenever $0 < |x - a| < \delta$, it follows that $|f(x) - L| < \epsilon$.

Read this as a challenge-and-response game. An adversary names an error tolerance $\epsilon > 0$ (how close to $L$ we must land). You must respond with a radius $\delta > 0$ so that every input within $\delta$ of $a$ (excluding $a$ itself) produces an output within $\epsilon$ of $L$. If you can always answer the challenge, the limit holds.

Proof sketch (linear limit): Show $\lim_{x\to 3}(2x - 1) = 5$. Given $\epsilon > 0$, we need $|(2x-1) - 5| < \epsilon$, i.e. $|2x - 6| = 2|x - 3| < \epsilon$, i.e. $|x - 3| < \tfrac{\epsilon}{2}$. Choosing $\delta = \tfrac{\epsilon}{2}$ works: if $0 < |x - 3| < \delta$ then $|(2x-1)-5| = 2|x-3| < 2\delta = \epsilon$.

Proof sketch (reciprocal limit): Show $\lim_{x\to 2}\frac{1}{x} = \frac{1}{2}$. Given $\epsilon > 0$, restrict attention to $|x - 2| < 1$, so $1 < x < 3$ and $|x| > 1$. Then $\left|\frac{1}{x} - \frac{1}{2}\right| = \frac{|x - 2|}{2|x|} < \frac{|x - 2|}{2}.$ Choosing $\delta = \min\{1,\,2\epsilon\}$ ensures $|x - 2| < \delta$ implies $\left|\frac{1}{x} - \frac{1}{2}\right| < \epsilon$.

2.3 One-Sided Limits

The left-hand limit $\lim_{x\to a^-} f(x) = L$ considers only $x < a$; the right-hand limit $\lim_{x\to a^+} f(x) = L$ considers only $x > a$. The two-sided limit exists if and only if both one-sided limits exist and are equal: $\lim_{x\to a} f(x) = L \iff \lim_{x\to a^-} f(x) = L = \lim_{x\to a^+} f(x).$ This is the standard tool for piecewise functions and for detecting jump discontinuities. The signum function $\operatorname{sgn}(x)$ has $\lim_{x\to 0^-}\operatorname{sgn}(x) = -1$ and $\lim_{x\to 0^+}\operatorname{sgn}(x) = 1$, so the two-sided limit at $0$ does not exist.

2.4 Limit Laws and Composition

If $\lim_{x\to a} f(x)$ and $\lim_{x\to a} g(x)$ both exist, then limits respect arithmetic:

• $\lim (f \pm g) = \lim f \pm \lim g$
• $\lim (f \cdot g) = \lim f \cdot \lim g$
• $\lim \frac{f}{g} = \frac{\lim f}{\lim g}$ provided $\lim g \neq 0$
• $\lim (cf) = c \lim f$ for a constant $c$
• $\lim [f(x)]^n = [\lim f(x)]^n$

Proof sketch (sum law): If $\lim f = L$ and $\lim g = M$, given $\epsilon > 0$ choose $\delta$ so $|f(x)-L| < \epsilon/2$ and $|g(x)-M| < \epsilon/2$ whenever $0 < |x-a| < \delta$. Then $|f(x)+g(x)-(L+M)| \le |f(x)-L| + |g(x)-M| < \epsilon$.

If $g$ is continuous at $L$ and $\lim_{x\to a} f(x) = L$, then $\lim_{x\to a} g(f(x)) = g(L)$. This composition law justifies evaluating limits of composite functions by substitution when the inner limit lands inside the domain of continuity of the outer function.

2.5 The Squeeze Theorem

If $g(x) \le f(x) \le h(x)$ near $a$ (except possibly at $a$) and $\lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L$, then $\lim_{x\to a} f(x) = L$.

Proof sketch: Given $\epsilon > 0$, choose $\delta$ so $|g(x)-L| < \epsilon$ and $|h(x)-L| < \epsilon$ for $0 < |x-a| < \delta$. Then $L - \epsilon < g(x) \le f(x) \le h(x) < L + \epsilon$, so $|f(x) - L| < \epsilon$.

This is the standard route to the foundational trigonometric limit $\lim_{x\to 0} \frac{\sin x}{x} = 1,$ derived geometrically by trapping $\frac{\sin x}{x}$ between $\cos x$ and $1$ for small $|x|$. A close companion is $\lim_{x\to 0} \frac{1 - \cos x}{x} = 0$.

2.6 Limits at Infinity and Infinite Limits

Limits at infinity describe end behavior: $\lim_{x\to\infty} f(x) = L$ means $f(x)$ can be kept within $\epsilon$ of $L$ by taking $x$ sufficiently large. Infinite limits such as $\lim_{x\to a} f(x) = \infty$ mean outputs grow without bound as $x$ approaches $a$. A vertical asymptote at $x = a$ occurs when at least one one-sided limit is $\pm\infty$; a horizontal asymptote $y = L$ occurs when $\lim_{x\to\pm\infty} f(x) = L$.

For rational functions $\frac{p(x)}{q(x)}$, compare degrees: if $\deg p < \deg q$ the horizontal asymptote is $y = 0$; if equal, $y$ equals the ratio of leading coefficients; if $\deg p > \deg q$ there is no horizontal asymptote (the limit is $\pm\infty$).

2.7 Continuity and Types of Discontinuity

A function $f$ is continuous at $a$ when three conditions all hold:

1. $f(a)$ is defined,
2. $\lim_{x\to a} f(x)$ exists, and
3. $\lim_{x\to a} f(x) = f(a)$.

If any condition fails, $f$ has a discontinuity at $a$:

• Removable — the limit exists but disagrees with (or is missing) the value; a "hole" you could patch.
• Jump — the one-sided limits exist but differ.
• Infinite — at least one one-sided limit is $\pm\infty$.

$f$ is continuous on an interval if it is continuous at every interior point; at endpoints we require only the appropriate one-sided continuity. Sums, products, quotients (with nonzero denominator), and compositions of continuous functions are continuous on their domains.

2.8 The Intermediate Value Theorem (IVT)

Statement: If $f$ is continuous on $[a,b]$ and $N$ is any value strictly between $f(a)$ and $f(b)$, then there exists $c \in (a,b)$ with $f(c) = N$.

Proof sketch (bisection idea): Suppose $f(a) < N < f(b)$. Bisect $[a,b]$ at its midpoint $m$. If $f(m) = N$ we are done; otherwise one half still has $f$ at endpoints straddling $N$. Repeat, producing nested intervals whose lengths $\to 0$. Continuity forces the common intersection point $c$ to satisfy $f(c) = N$.

The IVT guarantees roots: if $f$ changes sign on $[a,b]$, it has a zero in between. It does not guarantee uniqueness.

2.9 The Extreme Value Theorem (EVT)

Statement: If $f$ is continuous on a closed, bounded interval $[a,b]$, then $f$ attains an absolute maximum and an absolute minimum on $[a,b]$.

This underpins all optimization on closed intervals. Without continuity or a closed bounded domain, extrema may fail to exist (e.g. $f(x) = x$ on $(0,1)$ has no minimum).

2.10 L'Hôpital's Rule

When a limit has the indeterminate form $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$, and $f', g'$ exist near $a$ with $g' \neq 0$, then $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)},$ provided the right-hand limit exists or is $\pm\infty$. Differentiate numerator and denominator separately — not via the quotient rule. Forms $0\cdot\infty$, $\infty - \infty$, $1^\infty$, $0^0$, and $\infty^0$ must be rewritten (often via logarithms) before L'Hôpital applies.

2.11 Sequential Characterization (Optional Rigor)

For functions defined near $a$, $\lim_{x\to a} f(x) = L$ if and only if for every sequence $x_n \to a$ with $x_n \neq a$, we have $f(x_n) \to L$. This bridges limits and sequences and is useful for proving that a limit does not exist: find two sequences approaching $a$ with different limit values.

Example: $\lim_{x\to 0}\sin\frac{1}{x}$ does not exist: $x_n = \frac{1}{2\pi n} \Rightarrow \sin\frac{1}{x_n} = 0$, but $x_n' = \frac{1}{2\pi n + \pi/2} \Rightarrow \sin\frac{1}{x_n'} = 1$.

2.12 Growth Rates at Infinity

When $x \to \infty$, exponentials beat polynomials, which beat logarithms: $\ln x \ll x^p \ll a^x \ll x! \quad (p > 0,\; a > 1).$ This hierarchy resolves many $\tfrac{\infty}{\infty}$ limits without L'Hôpital. For example, $\lim_{x\to\infty}\frac{x^{100}}{e^x} = 0$ because exponential growth dominates any power.

2.13 Continuity of Compositions and Piecewise Functions

If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f \circ g$ is continuous at $a$. For a piecewise function $f(x) = \begin{cases} p(x), & x < a \\ q(x), & x \ge a \end{cases}$, continuity at $a$ requires $\lim_{x\to a^-} p(x) = \lim_{x\to a^+} q(x) = f(a)$.

2.14 Limits and Asymptotic Notation

We write $f(x) \sim g(x)$ as $x \to a$ when $\lim_{x\to a}\frac{f(x)}{g(x)} = 1$. For instance, $\sin x \sim x$ as $x \to 0$ and $\ln x \sim x - 1$ as $x \to 1$. Asymptotic equivalence simplifies limit calculations by replacing complicated expressions with simpler ones of the same local behavior.