Comprehensive Calculus Study Guide: Continuity, One-Sided Limits, Intermediate Value Theorem, and Infinite Limits

Formal Definition and Conditions of Continuity

• Overview of Continuity Concepts in Calculus:

  • Elementary definitions of continuity, such as the ability to draw a graph without lifting a pencil, are insufficient for rigorous calculus.
  • Statements encountered in algebra—such as claims that the sine function only produces values between $-1$ and $1$, or that logarithms can only accept positive inputs—are context-dependent simplifications that rely on restricting domain analysis strictly to real numbers rather than complex numbers.
  • In real-valued calculus, precise, formal definitions are required to establish the properties of continuous functions.

• Formal Definition of Continuity at a Point:

  • Let $f$ be a function, and let $c$ be a real number.
  • The function $f$ is defined to be continuous at the number $c$ if and only if the following three conditions are simultaneously met:
  • Condition 1: $f(c)$ exists. The number $c$ must be in the domain of $f$, producing a defined real output.
  • Condition 2: $\lim_{x \to c} f(x)$ exists. As $x$ approaches $c$ from both the left and right sides, the function values must approach a single finite real number.
  • Condition 3: $\lim_{x \to c} f(x) = f(c)$. The limit as $x$ approaches $c$ must equal the value obtained by directly evaluating the function at $c$.

• Discontinuity at a Point:

  • If any single condition of the three fails to hold, then $f$ is not continuous at $c$ (also described as $f$ being discontinuous at $c$).

• Definition of Continuity on an Interval:

  • If a function $f$ is continuous at every number $x$ within an interval $I$, then $f$ is continuous on the interval $I$.

Practical Strategy for Continuity and Graphical Failure Modes

• Working Strategy for Continuity Problems:

  • From a practical problem-solving perspective, assume that a given function is continuous for all real numbers across $(-\infty, \infty)$ and then actively search for specific exceptions or domain restrictions that break continuity.

• Graphical Manifestations of Continuity Failures:

  • Failure of Condition 1 ($f(c)$ does not exist):
  • Visual Representation: A hole exists in the graph at $x = c$.
  • Behavior: The left-hand limit and right-hand limit approach the exact same value (meaning $\lim_{x \to c} f(x)$ exists). However, evaluating the function at $x = c$ is impossible because no point exists there.
  • Consequence: Failure of Condition 1 automatically causes Condition 3 to fail as well.
  • Failure of Condition 2 ($\lim_{x \to c} f(x)$ does not exist):
  • Visual Representation: A jump, break, or disconnect exists in the graph at $x = c$.
  • Behavior: The function value $f(c)$ may be defined at a specific point, satisfying Condition 1. However, the path approached from the left side of $c$ does not align with the path approached from the right side of $c$.
  • Consequence: Because the left-hand and right-hand limits differ, the two-sided limit $\lim_{x \to c} f(x)$ does not exist, violating Condition 2.
  • Failure of Condition 3 ($\lim_{x \to c} f(x) \neq f(c)$):
  • Visual Representation: The graph has a continuous path with a hole at $x = c$, but the actual point at $x = c$ is plotted at a completely different height above or below the hole.
  • Behavior: $f(c)$ exists at the displaced point (satisfying Condition 1). The two-sided limit $\lim_{x \to c} f(x)$ exists along the curve (satisfying Condition 2).
  • Consequence: Because the finite value of $\lim_{x \to c} f(x)$ does not equal the finite value of $f(c)$, Condition 3 fails.

Domain Exceptions in Real-Valued Calculus

• Context of Real vs. Complex Analysis:

  • In complex analysis (calculus extended to complex numbers), operations such as taking square roots of negative numbers or evaluating logarithms of negative values are defined. Complex analysis is studied in upper-level mathematics, computer science, or senior electrical engineering courses.
  • In single-variable calculus, operations are restricted strictly to real numbers. An expression is categorized as undefined if it does not produce a real number.

• Primary Domain Exceptions in Real Numbers:

  • Division by zero: Any expression with a denominator equal to $0$ is strictly undefined.
  • Even roots of negative numbers: Even-indexed roots (such as square roots $\sqrt{x}$, fourth roots $\sqrt[4]{x}$, or sixth roots $\sqrt[6]{x}$) of negative numbers are undefined in real numbers. Odd-indexed roots (such as cube roots $\sqrt[3]{x}$) of negative numbers are defined.
  • Logarithms of non-positive numbers: The expression $\log_b(x)$ is undefined in real numbers for $x \le 0$
  • Inverse trigonometric functions outside domain limits: Inverse trig functions (such as $\arcsin(x)$ or $\arccos(x)$) are undefined outside their real domains ($[-1, 1]$).

• Continuous Function Families:

  • Polynomials: All polynomial functions are continuous for all real numbers on the interval $(-\infty, \infty)$.
  • Sine and Cosine: The functions $\sin(x)$ and $\cos(x)$ are continuous for all real numbers on $(-\infty, \infty)$. Other trigonometric functions ($\tan(x)$, $\cot(x)$, $\sec(x)$, $\csc(x)$) have periodic discontinuities.
  • Quotients of continuous functions: A quotient of two continuous functions $\frac{p(x)}{q(x)}$ is continuous everywhere except where the denominator $q(x) = 0$

Comprehensive Examples of Continuity Analysis

• Discussion of Continuity Defined:

  • Instructions asking to "discuss the continuity of a function" require explicitly stating the exact intervals on which the function is continuous, or identifying the exact points where the function is discontinuous.

• Example 1: Rational Function $f(x) = \frac{1}{x}$

  • Analysis: Assume continuity across $(-\infty, \infty)$ and search for exceptions. The variable in the denominator causes division by zero when $x = 0$.
  • Evaluation: $f(0)$ is undefined, violating Condition 1.
  • Conclusion: $f(x)$ is continuous on $(-\infty, 0) \cup (0, \infty)$, or equivalently, $f(x)$ is discontinuous at $x = 0$

• Example 2: Piecewise Polynomial Function $g(x) = \begin{cases} x + 1 & \text{if } x \le 0 \\ x^2 + 1 & \text{if } x > 0 \end{cases}$

  • Analysis for $x < 0$: $x + 1$ is a polynomial, so $g(x)$ is continuous on $(-\infty, 0)$.
  • Analysis for $x > 0$: $x^2 + 1$ is a polynomial, so $g(x)$ is continuous on $(0, \infty)$.
  • Analysis at Boundary Point $x = 0$:
  • Condition 1: $g(0) = 0 + 1 = 1$ (exists).
  • Condition 2: Compute one-sided limits by direct substitution into the polynomial components:
    • Left-hand limit: $\lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} (x + 1) = 0 + 1 = 1$
    • Right-hand limit: $\lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (x^2 + 1) = 0^2 + 1 = 1$
    • Because both one-sided limits equal $1$, the two-sided limit exists: $\lim_{x \to 0} g(x) = 1$
  • Condition 3: $\lim_{x \to 0} g(x) = 1 = g(0)$.
  • Conclusion: $g(x)$ is continuous at $x = 0$. Therefore, $g(x)$ is continuous on $(-\infty, \infty)$ (never discontinuous).

• Example 3: Finding Parameter $a$ for Global Continuity

  • Problem Statement: Find a number $a$ such that the function $f(x) = \begin{cases} x^3 & \text{if } x \le 2 \\ a x^2 & \text{if } x > 2 \end{cases}$ is continuous for all real numbers.
  • Analysis for $x \neq 2$: Both $x^3$ and $a x^2$ are polynomials, ensuring continuity on $(-\infty, 2)$ and $(2, \infty)$.
  • Boundary Conditions at $x = 2$:
  • Function Value: $f(2) = 2^3 = 8$
  • Left-hand Limit: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^3 = 2^3 = 8$
  • Right-hand Limit: $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (a x^2) = a(2)^2 = 4a$
  • Continuity Requirement: Equate the right-hand limit to the left-hand limit and function value:     $4a = 8$$a = 2$
  • Conclusion: Setting $a = 2$ guarantees continuity at $x = 2$, making $f(x)$ continuous on $(-\infty, \infty)$.

• Example 4: Rational Function with Multiple Discontinuities

  • Problem Statement: Discuss the continuity of $f(x) = \frac{x}{x^2 - 4}$.
  • Analysis: Both numerator and denominator are polynomials. Discontinuities occur solely where the denominator vanishes:     $x^2 - 4 = 0$$(x - 2)(x + 2) = 0$$x = 2 \quad \text{or} \quad x = -2$
  • Conclusion: $f(x)$ is continuous on $(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$, and discontinuous at $x = -2$ and $x = 2$

• Example 5: Non-Continuable Function involving Trigonometric Limits

  • Problem Statement: Find a number $a$ such that $g(x) = \begin{cases} \frac{4 \sin(x)}{x} & \text{if } x < 0 \\ a x + 5 & \text{if } x \ge 0 \end{cases}$ is continuous on $(-\infty, \infty)$.
  • Essential Identity: The fundamental trigonometric limit derived via the Squeeze Theorem is:     $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$
  • Evaluation at $x = 0$:
  • Function Value: $g(0) = a(0) + 5 = 5$
  • Left-hand Limit: Applying limit linearity properties,       $\lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} \frac{4 \sin(x)}{x} = 4 \cdot \lim_{x \to 0^-} \frac{\sin(x)}{x} = 4(1) = 4$
  • Right-hand Limit: $\lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (a x + 5) = a(0) + 5 = 5$
  • Analysis of Continuity Conditions: The left-hand limit as $x \to 0^-$ is fixed at $4$. However, the function value $g(0)$ is fixed at $5$. Because the left-hand limit ($4$) does not equal $g(0)$ ($5$), Condition 3 fails regardless of the choice of $a$
  • Conclusion: No value of $a$ exists for which $g(x)$ is continuous at $x = 0$. It is impossible for $g(x)$ to be continuous on $(-\infty, \infty)$.

One-Sided Limits and Boundary Behavior

• Mathematical Notation for One-Sided Limits:

  • Right-Hand Limit (Limit from the Right):
  • Notation: $\lim_{x \to c^+} f(x)$
  • Reading: "The limit as $x$ approaches $c$ from the right."
  • Meaning: The value approached by $f(x)$ as $x$ approaches $c$ through values strictly greater than $c$ ($x > c$). The superscript plus sign represents directional approach from the positive side.
  • Left-Hand Limit (Limit from the Left):
  • Notation: $\lim_{x \to c^-} f(x)$
  • Reading: "The limit as $x$ approaches $c$ from the left."
  • Meaning: The value approached by $f(x)$ as $x$ approaches $c$ through values strictly less than $c$ ($x < c$). The superscript minus sign represents directional approach from the negative side.

• Fundamental Two-Sided Limit Theorem:

  • A two-sided limit $\lim_{x \to c} f(x) = L$ exists if and only if both one-sided limits exist and are equal to $L$$\lim_{x \to c^+} f(x) = L \quad \text{and} \quad \lim_{x \to c^-} f(x) = L \iff \lim_{x \to c} f(x) = L$
  • If $\lim_{x \to c^+} f(x) \neq \lim_{x \to c^-} f(x)$, the two-sided limit $\lim_{x \to c} f(x)$ does not exist (DNE).

• Analysis of Domain Boundaries (Upper Semicircle Example):

  • Consider the function $f(x) = \sqrt{4 - x^2}$.
  • Algebraic Derivation: Setting $y = \sqrt{4 - x^2}$ and squaring both sides gives $y^2 = 4 - x^2 \implies x^2 + y^2 = 4$ with $y \ge 0$. This represents the upper half of a circle centered at $(0, 0)$ with radius $2$, defined on the closed interval $[-2, 2]$.
  • Limit at Interior Point $x = 0$:     $\lim_{x \to 0} \sqrt{4 - x^2} = 2$     Both left and right approaches yield $2$, so the two-sided limit is $2$
  • Limits at Endpoint $x = 2$:
  • Left-hand limit: Approaching $x = 2$ from values inside the domain ($x < 2$):       $\lim_{x \to 2^-} \sqrt{4 - x^2} = 0$
  • Right-hand limit: For $x > 2$, $4 - x^2 < 0$, making $\sqrt{4 - x^2}$ undefined in real numbers:       $\lim_{x \to 2^+} \sqrt{4 - x^2} \quad \text{does not exist (undefined)}$
  • Two-sided limit: Because the right-hand limit does not exist, the two-sided limit $\lim_{x \to 2} \sqrt{4 - x^2}$ does not exist.

• Piecewise One-Sided Limit Evaluation:

  • Let $f(x) = \begin{cases} 3x + 5 & \text{if } x < 0 \\ x^2 - 1 & \text{if } x > 0 \\ 5 & \text{if } x = 0 \end{cases}$
  • Value at origin: $f(0) = 5$
  • Left-hand limit ($x \to 0^-$):     $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (3x + 5) = 3(0) + 5 = 5$
  • Right-hand limit ($x \to 0^+$):     $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x^2 - 1) = 0^2 - 1 = -1$
  • Two-Sided Limit Evaluation: Because $\lim_{x \to 0^-} f(x) = 5$ and $\lim_{x \to 0^+} f(x) = -1$ are unequal, $\lim_{x \to 0} f(x)$ does not exist (DNE).
  • Continuity Assessment: The function is discontinuous at $x = 0$ due to failure of Condition 2.

The Intermediate Value Theorem

• Categorization of Mathematical Theorems:

  • Constructive Theorems (Formulas): Theorems that provide an explicit computational formula to calculate an exact result (e.g., the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for roots of $a x^2 + b x + c = 0$).
  • Theorems of Existence: Theorems that state and prove that a value or solution exists within a specified set, without providing a direct formula to compute it.
  • Role of Existence Theorems: Existence theorems guarantee that mathematical entities exist, ensuring that attempts to solve equations or construct numerical algorithms are mathematically valid. An example in algebra is the Fundamental Theorem of Algebra, which guarantees that every non-constant polynomial has at least one zero in the complex numbers.

• Formal Statement of the Intermediate Value Theorem (IVT):

  • Let $f$ be a function that is continuous on the closed interval $[a, b]$.
  • Suppose $f(a) \neq f(b)$.
  • Let $k$ be any real number strictly between $f(a)$ and $f(b)$.
  • Then there exists at least one number $c$ in the open interval $(a, b)$ such that:     $f(c) = k$

• Geometric Interpretation of IVT:

  • If a continuous function connects the point $(a, f(a))$ to the point $(b, f(b))$ on a Cartesian plane, the curve must cross every horizontal line $y = k$ positioned between $f(a)$ and $f(b)$ at least once at some horizontal coordinate $x = c$ in $(a, b)$.

• Proof Characteristics:

  • Proofs of existence theorems require rigorous real analysis properties (such as completeness of real numbers) and are omitted in introductory calculus due to advanced structural complexity.

Infinite Limits and Vertical Asymptotes

• Behavior of Reciprocal Functions near $x = 0$:

  • Analysis of $f(x) = \frac{1}{x}$:

  • Table of evaluations as $x \to 0$:

    • $x = 1 \implies f(1) = 1$
    • $x = -1 \implies f(-1) = -1$
    • $x = \frac{1}{2} \implies f\left(\frac{1}{2}\right) = 2$
    • $x = -\frac{1}{2} \implies f\left(-\frac{1}{2}\right) = -2$
    • $x = \frac{1}{10} \implies f\left(\frac{1}{10}\right) = 10$
    • $x = -\frac{1}{10} \implies f\left(-\frac{1}{10}\right) = -10$
    • $x = \frac{1}{1000} \implies f\left(\frac{1}{1000}\right) = 1000$
    • $x = -\frac{1}{1000} \implies f\left(-\frac{1}{1000}\right) = -1000$

  • Observation: As $x \to 0^+$, $f(x) \to +\infty$. As $x \to 0^-$, $f(x) \to -\infty$.

  • Limit Conclusion: Because the left-hand and right-hand behaviors grow in opposite directions, the two-sided limit $\lim_{x \to 0} \frac{1}{x}$ does not exist (DNE).

  • Analysis of $g(x) = \frac{1}{x^2}$:

  • Table of evaluations as $x \to 0$:

    • $x = \pm 1 \implies g(\pm 1) = 1$
    • $x = \pm \frac{1}{2} \implies g\left(\pm \frac{1}{2}\right) = 4$
    • $x = \pm \frac{1}{10} \implies g\left(\pm \frac{1}{10}\right) = 100$
    • $x = \pm \frac{1}{1000} \implies g\left(\pm \frac{1}{1000}\right) = 1000000$

  • Observation: As $x$ approaches $0$ from both left and right sides, $g(x)$ increases without bound in the positive direction.

  • Limit Notation: Strictly speaking, because infinity is not a real number, the limit does not exist as a real finite value. However, because both sides exhibit identical unbounded growth, this specific behavior is denoted using infinite limit notation:       $\lim_{x \to 0} \frac{1}{x^2} = \infty$

  • Meaning of Equal Sign with Infinity: Writing $= \infty$ is a notation convention expressing unbounded growth rather than literal numerical equality, as infinity represents an unbounded process rather than a static real number.

• Formal Precise Definition of an Infinite Limit ($\lim_{x \to c} f(x) = \infty$):

  • Let $f$ be a function defined on an open interval containing $c$, except possibly at $c$ itself.
  • The statement $\lim_{x \to c} f(x) = \infty$ means that for every positive real number $M > 0$, there exists a corresponding positive real number $\delta > 0$ such that for all $x$:     $\text{if } 0 < |x - c| < \delta, \quad \text{then } f(x) > M$

• Geometric Meaning of Infinite Limit Definition:

  • For any horizontal boundary line $y = M$ chosen arbitrarily high above the x-axis, there exists a sufficiently small distance $\delta$ from $c$ such that all points $x$ within $(c - \delta, c + \delta)$ (excluding $x = c$) yield function values $f(x)$ positioned strictly above $M$
  • This behavior mathematically defines a vertical asymptote where both sides of the graph curve upward toward positive infinity as $x$ approaches $c$.