∫Calculus I Unit 2 – Limits

What Are Limits?

• Limits describe the behavior of a function as the input approaches a certain value
• Used to determine the value of a function at a point where it may be undefined or indeterminate
• Denoted by $\lim_{x \to a} f(x) = L$, meaning as $x$ approaches $a$, the function $f(x)$ approaches the value $L$
• Helps analyze functions near critical points, such as discontinuities or points of non-differentiability
• Essential concept in calculus, forming the foundation for derivatives and integrals
• Allow for the study of asymptotic behavior and the convergence or divergence of functions
• Provide a way to describe the behavior of a function near a point without requiring the function to be defined at that point

Key Concepts and Definitions

• Limit: The value a function approaches as the input approaches a specific value
• One-sided limits: Limits that consider the function's behavior from only one direction (left or right)
  • Left-hand limit: $\lim_{x \to a^-} f(x)$, approaching $a$ from values less than $a$
  • Right-hand limit: $\lim_{x \to a^+} f(x)$, approaching $a$ from values greater than $a$
• Two-sided limit: A limit that considers the function's behavior from both directions
• Limit laws: Rules that allow for the manipulation and simplification of limits
  • Sum, difference, product, and quotient rules for limits
  • Power, root, and constant multiple rules for limits
• Indeterminate forms: Expressions that arise when evaluating limits, requiring further investigation (0/0, $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$)
• Squeeze theorem: If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$

Types of Limits

• Finite limits: Limits that approach a real number as $x$ approaches a specific value
• Infinite limits: Limits that approach positive or negative infinity as $x$ approaches a specific value
  • Vertical asymptotes: Occur when a function approaches infinity as $x$ approaches a finite value
• Limits at infinity: Describe the behavior of a function as $x$ approaches positive or negative infinity
  • Horizontal asymptotes: Occur when a function approaches a finite value as $x$ approaches positive or negative infinity
• One-sided limits: Limits that consider the function's behavior from only one direction (left or right)
• Limits of piecewise functions: Require examining the limit of each piece separately and considering the value of the function at the point of interest
• Limits of trigonometric functions: Often involve using trigonometric identities and the squeeze theorem
• Limits of exponential and logarithmic functions: Utilize properties of exponents and logarithms to simplify and evaluate limits

Techniques for Finding Limits

• Direct substitution: Replacing the variable with the limit value and simplifying
• Factoring and canceling: Factoring the numerator and denominator to cancel common factors and simplify the limit
• Rationalizing the numerator or denominator: Multiplying by the conjugate to eliminate radicals in the denominator
• Using trigonometric identities: Applying trigonometric identities to simplify limits involving trigonometric functions
• L'Hôpital's rule: For indeterminate forms of type $0/0$ or $\infty/\infty$, differentiate the numerator and denominator separately and evaluate the limit of the resulting quotient
  • Can be applied repeatedly if the indeterminate form persists after the first application
• Squeeze theorem: Bounding the function between two other functions with known limits to determine the limit of the original function
• Limit laws: Applying the sum, difference, product, quotient, power, root, and constant multiple rules to simplify limits
• Graphical approach: Estimating the limit by examining the graph of the function near the point of interest

Common Limit Problems

• Limits involving absolute value functions: Require considering the piecewise nature of the function and evaluating one-sided limits
• Limits of rational functions: Often lead to indeterminate forms (0/0 or $\infty/\infty$) and require factoring, canceling, or applying L'Hôpital's rule
• Limits involving radicals: May require rationalizing the numerator or denominator to simplify the limit
• Limits of piecewise functions: Require examining the limit of each piece separately and considering the value of the function at the point of interest
• Limits involving trigonometric functions: Often involve using trigonometric identities and the squeeze theorem to simplify and evaluate the limit
• Limits of exponential and logarithmic functions: Utilize properties of exponents and logarithms to simplify and evaluate limits
• Limits involving indeterminate forms: Require further investigation using techniques such as factoring, L'Hôpital's rule, or the squeeze theorem

Applications of Limits

• Defining derivatives: The derivative of a function $f(x)$ at a point $a$ is defined as the limit of the difference quotient as $h$ approaches 0
  • $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
• Defining integrals: The definite integral of a function $f(x)$ over the interval $[a, b]$ is defined as the limit of Riemann sums as the number of subintervals approaches infinity
  • $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$
• Analyzing the behavior of functions: Limits help determine the behavior of functions near critical points, such as discontinuities or points of non-differentiability
• Approximating solutions to equations: Limits can be used in numerical methods, such as Newton's method, to approximate roots of equations
• Evaluating improper integrals: Limits are used to assign values to integrals with infinite intervals or unbounded integrands
• Studying convergence and divergence of sequences and series: Limits help determine whether a sequence or series converges to a specific value or diverges

Continuity and Limits

• Continuity: A function is continuous at a point $a$ if $\lim_{x \to a} f(x) = f(a)$
  • The function must be defined at $a$, the limit must exist, and the limit must equal the function value at $a$
• Types of discontinuities:
  • Removable discontinuity: The limit exists, but the function is not defined at the point or does not equal the limit value
  • Jump discontinuity: The left and right-hand limits exist but are not equal
  • Infinite discontinuity: The limit does not exist because the function approaches positive or negative infinity
• Intermediate Value Theorem: If $f(x)$ is continuous on the closed interval $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists a $c$ in $[a, b]$ such that $f(c) = k$
• Extreme Value Theorem: If $f(x)$ is continuous on a closed interval $[a, b]$, then $f(x)$ attains both a maximum and minimum value on the interval
• Continuity and differentiability: If a function is differentiable at a point, it must be continuous at that point, but the converse is not always true

Practice and Review

• Evaluate limits using direct substitution, factoring, and canceling
• Find limits of piecewise functions by examining each piece separately
• Apply L'Hôpital's rule to evaluate limits involving indeterminate forms
• Use the squeeze theorem to find limits of functions bounded by other functions with known limits
• Determine the continuity of functions at specific points and classify discontinuities
• Solve problems involving the Intermediate Value Theorem and the Extreme Value Theorem
• Analyze the behavior of functions using limits, including finding asymptotes and points of non-differentiability
• Apply limits to define derivatives and integrals, and solve related problems
• Practice evaluating limits of various types of functions, including rational, trigonometric, exponential, and logarithmic functions