Limits and Continuity

What You Will Learn in This Chapter

Section	Description
2.1 The Concept of a Limit	What it means for a function to approach a value as $x$ approaches a point
2.2 One-Sided Limits	Left-hand and right-hand limits; when the two-sided limit exists
2.3 Infinite Limits	Limits that grow without bound; vertical behavior near singularities
2.4 Limits at Infinity	How functions behave as $x\to \pm \mathrm{\infty }$; horizontal asymptotes
2.5 Special Limits	A reference list of 17 essential limits involving powers, exponentials, logs, and trig
2.6 Theorems for Calculating Limits	Limit laws, indeterminate forms, the Sandwich Theorem, bounded-function theorem
2.7 The Limit of sin(x)/x as x→0	Geometric proof and applications including tan(x)/x and (1−cos x)/x²
2.8 Continuity	Definition, types of discontinuity, elementary continuous functions, algebraic operations
2.9 How to Evaluate Limits	Factoring, rationalizing, common denominator, and leading-term techniques
2.10 Asymptotes	Vertical, horizontal, and oblique asymptotes with examples
2.11 Exercises	Practice problems covering all topics in the chapter

Why Limits Matter

Calculus has two major branches: differential calculus (rates of change and tangent lines) and integral calculus (areas and accumulated change). Both are defined entirely in terms of limits.

The notion of a limit captures the idea of a function getting arbitrarily close to a value, without necessarily reaching it. For example, the function $f(x)={\displaystyle \frac{2x^2-2}{x-1}}$ is not defined at $x=1$, yet its values approach $4$ as $x$ gets close to $1$. That approach is the limit.

Real-World Applications

• Physics: instantaneous velocity is the limit of average velocity as the time interval shrinks to zero
• Engineering: stress analysis requires understanding behavior near singularities (infinite limits)
• Economics: marginal cost and revenue are defined as limits of difference quotients
• Biology: population models use limits at infinity to describe long-run behavior

Frequently Asked Questions

What is the difference between a limit and a function value?

The **limit** $\underset{x\to a}{lim}f(x)$ describes what $f(x)$ approaches as $x$ gets close to $a$, it does not depend on $f(a)$ at all. The function **value** $f(a)$ is simply what $f$ equals at $x=a$. A limit can exist even if $f(a)$ is undefined or has a different value than the limit.

When does a limit fail to exist?

A two-sided limit $\underset{x\to a}{lim}f(x)$ fails to exist in three common situations: (1) the left-hand and right-hand limits are not equal (jump discontinuity), (2) $f(x)$ grows without bound as $x\to a$ (infinite limit), or (3) $f(x)$ oscillates without settling near any value (oscillating discontinuity).

What does it mean for a function to be continuous?

A function $f$ is **continuous** at $a$ if two conditions hold: $f(a)$ is defined, and $\underset{x\to a}{lim}f(x)=f(a)$. Intuitively, the graph has no holes, jumps, or vertical breaks at $x=a$. Continuous functions can be drawn without lifting the pen.

What is an indeterminate form?

An **indeterminate form** arises when the limit laws cannot directly tell you the value of a limit. The four basic indeterminate forms are $\frac{0}{0}$, $\frac{\pm \mathrm{\infty }}{\pm \mathrm{\infty }}$, $0\cdot (\pm \mathrm{\infty })$, and $\mathrm{\infty }-\mathrm{\infty }$. Each requires additional algebraic work (factoring, rationalizing, etc.) or L'Hôpital's Rule to resolve.

Why is the limit sin(x)/x as x→0 equal to 1?

When $x$ is measured in **radians**, the arc length $x$ and the chord $\sin x$ are nearly equal for small angles. More precisely, a geometric area argument shows for $x$ near $0$, and since $\cos x\to 1$, the Sandwich Theorem forces the ratio to equal $1$.