Functions, Limits, and Continuity - Theory & Concepts

Functions, Limits, and Continuity

Differential calculus begins with a rigorous understanding of functions and their behavior as inputs approach specific values. This topic covers the essential building blocks: function properties, limits, the Squeeze Theorem, and continuity, which are prerequisites for understanding the derivative.

Basic Function Properties

Before exploring limits, it is crucial to understand the foundational properties of functions, which often simplify calculus operations and graphing. Two of the most important concepts are the domain and range.

Domain and Range

Domain: The set of all possible input values (usually $x$ ) for which the function is defined and produces a valid real number. Common restrictions include avoiding division by zero and preventing even roots of negative numbers.
Range: The set of all possible output values (usually $y$ or $f(x)$ ) that the function can produce given its domain.

Special Types of Functions

Absolute Value Function: $f(x) = |x|$ . It represents the distance from zero. It is defined piecewise as $f(x) = x$ if $x \ge 0$ , and $f(x) = -x$ if $x < 0$ . Its graph is V-shaped, with a sharp corner at the origin.
Piecewise Functions: Functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Limits and continuity must be carefully checked at the boundaries where the rules change.

Even, Odd, and Periodic Functions

Even Functions: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain. Its graph is symmetric with respect to the y-axis. (e.g., $f(x) = x^2, \cos x$ ).
Odd Functions: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain. Its graph is symmetric with respect to the origin. (e.g., $f(x) = x^3, \sin x$ ).
Periodic Functions: A function is periodic if there exists a positive constant $P$ such that $f(x + P) = f(x)$ for all $x$ . The smallest such $P$ is the fundamental period.

The Limit Concept

Limit of a Function

The limit of a function $f(x)$ as $x$ approaches a value $c$ , denoted as $\lim_{x \to c} f(x) = L$ , means that as $x$ gets arbitrarily close to $c$ (but not equal to $c$ ), the value of $f(x)$ gets arbitrarily close to $L$ .

Key Properties of Limits

Independence from Function Value: The value of $f(c)$ does not affect $\lim_{x \to c} f(x)$ . The function doesn't even need to be defined at $c$ .
Left-hand limit: $\lim_{x \to c^-} f(x)$ (approaching $c$ from values smaller than $c$ ).
Right-hand limit: $\lim_{x \to c^+} f(x)$ (approaching $c$ from values larger than $c$ ).
Existence: For the general limit $\lim_{x \to c} f(x)$ to exist, the left-hand and right-hand limits must be equal.

Interact with the simulation below to explore limits.

Limit Explorer

Approach x = 1

-13

x0.5000

f(x)1.5000

Target (L)2

Notice how f(x) gets closer to L as x approaches c, even if f(c) is undefined.

Formal Epsilon-Delta Definition

While the intuitive definition serves for most engineering applications, the formal definition of a limit ensures mathematical rigor. It states that

\lim_{x \to c} f(x) = L

if for every number

\epsilon > 0

(no matter how small), there exists a corresponding number

\delta > 0

such that if

0 < |x - c| < \delta

, then

|f(x) - L| < \epsilon

Interact with the simulation below to explore the formal Epsilon-Delta limit definition.

Epsilon-Delta Visualization

Adjust the output tolerance ( $\epsilon$ ) to see the required input window ( $\delta$ ). The function is $f(x) = 2x + 1$ with a limit of $L = 5$ at $c = 2$ . Notice that $\delta = \epsilon / 2$ .

Tolerance

\epsilon

: 1.00

Values:

Limit

L = 5

L + \epsilon

= 6.00

L - \epsilon

= 4.00

Input

c = 2

Required

\delta

= 0.50

c + \delta

= 2.50

c - \delta

= 1.50

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Limit Laws

Assuming

\lim_{x \to c} f(x) = L

and

\lim_{x \to c} g(x) = M

exist, the following laws allow us to compute limits algebraically.

Fundamental Limit Laws

STEP-BY-STEP

Sum/Difference: $\lim_{x \to c} [f(x) \pm g(x)] = L \pm M$
Product: $\lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M$
Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ (provided $M \neq 0$ )
Power: $\lim_{x \to c} [f(x)]^n = L^n$
Root: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$ (if $n$ is even, $L > 0$ )

The Squeeze (Sandwich) Theorem

The Squeeze Theorem is a powerful tool for evaluating the limits of functions that are difficult or impossible to evaluate directly, by "squeezing" them between two simpler functions whose limits are known.

Squeeze Theorem

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an open interval containing $c$ (except possibly at $c$ itself), and if:

Squeeze Theorem Limit

The resulting limit from the Squeeze Theorem.

\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L

Variables

Symbol	Description	Unit
$f (x)$	Lower bounding function	-
$h (x)$	Upper bounding function	-
$c$	The limit point	-
$L$	The limit value	-

Then it must be true that

\lim_{x \to c} g(x) = L

. This theorem is crucial for proving that

\lim_{x \to 0} \frac{\sin x}{x} = 1

Evaluating Limits

1. Direct Substitution

For continuous functions (polynomials, rational functions where denominator is not 0, trigonometric functions), we can often find the limit by simply plugging in the value.

2. Factoring Method (Indeterminate Forms)

When direct substitution results in

\frac{0}{0}

, we have an indeterminate form. This suggests a hole in the graph. We often need to factor and cancel common terms.

3. Conjugate Method

When dealing with radicals resulting in

\frac{0}{0}

, multiplying by the conjugate often helps to rationalize the numerator or denominator.

Continuity

Continuity is a central concept in calculus. Intuitively, a function is continuous if you can draw its graph without lifting your pen.

Continuity at a Point

A function $f(x)$ is continuous at a point $x = c$ if three conditions are met:

Defined: $f(c)$ is defined.
Limit Exists: $\lim_{x \to c} f(x)$ exists.
Matches: $\lim_{x \to c} f(x) = f(c)$ .

Discontinuity

If any of these three conditions fail, the function is discontinuous at $x=c$ .

Types of Discontinuity

Removable Discontinuity: The limit exists, but $f(c)$ is either undefined or not equal to the limit. (e.g., a hole in the graph).
Jump Discontinuity: The left-hand and right-hand limits exist but are not equal. (e.g., piecewise functions like step functions).
Infinite Discontinuity: One or both of the one-sided limits go to infinity. (e.g., vertical asymptotes).

Interact with the simulation below to explore various discontinuity types.

Continuity & Discontinuity Explorer

Function Type

Analysis at x = 2

The function is continuous at x = 2. The limit as x approaches 2 exists and is equal to the function value f(2) = 4.

Formula

f(x) = x^2

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The Intermediate Value Theorem (IVT)

The Intermediate Value Theorem is a fundamental property of continuous functions, guaranteeing that a continuous curve must pass through every intermediate y-value between two endpoints.

Intermediate Value Theorem

If a function $f$ is continuous on a closed interval $[a, b]$ , and $N$ is any number strictly between $f(a)$ and $f(b)$ , then there exists at least one number $c$ in the open interval $(a, b)$ such that $f(c) = N$ .

Application of IVT

A primary application of the IVT is proving the existence of roots (zeros). If $f$ is continuous on $[a, b]$ , and $f(a)$ and $f(b)$ have opposite signs, the IVT guarantees there is at least one point $c$ in $(a, b)$ where $f(c) = 0$ .

Infinite Limits and Limits at Infinity

Types of Limits at Infinity

Infinite Limits: If $f(x)$ increases or decreases without bound as $x \to c$ , we say the limit is $\infty$ or $-\infty$ . This indicates a vertical asymptote at $x = c$ .
Limits at Infinity: We analyze the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$ . This relates to horizontal asymptotes.

Asymptotes

Types of Asymptotes

Vertical Asymptote (VA): Occurs at $x = c$ if $\lim_{x \to c^+} f(x) = \pm\infty$ or $\lim_{x \to c^-} f(x) = \pm\infty$ . Typically found where the denominator of a rational function is zero and the numerator is non-zero.
Horizontal Asymptote (HA): Occurs at $y = L$ if $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ . Describes the end behavior of the function.
Oblique (Slant) Asymptote (OA): Occurs when the degree of the numerator is exactly one greater than the degree of the denominator. Found using polynomial long division. The quotient forms the equation of the slant asymptote $y = mx + b$ .

Key Takeaways

A limit describes the behavior of a function near a point, requiring both one-sided limits to be equal for existence.
The Squeeze Theorem evaluates difficult limits by sandwiching them between functions with known limits.
Continuity guarantees that the limit equals the function value. Discontinuities can be removable, jump, or infinite.
The Intermediate Value Theorem (IVT) ensures that a continuous function assumes every value between its endpoints, useful for proving the existence of roots.
Limits at infinity and infinite limits define the horizontal and vertical asymptotes of a function, respectively.

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