∬Differential Calculus Unit 2 – Limits and Their Properties

Key Concepts

• Limits describe the behavior of a function as the input approaches a certain value
• Limits are essential for understanding continuity, derivatives, and integrals in calculus
• One-sided limits (left-hand and right-hand limits) help analyze the behavior of a function from different directions
• Limits can be evaluated using various techniques such as direct substitution, factoring, and rationalization
• Indeterminate forms (0/0, ∞/∞, 0⋅∞, ∞-∞, 1^∞, ∞^0, 0^0) require special methods like L'Hôpital's Rule to evaluate the limit
• Continuity of a function at a point depends on the existence of the limit and its equality to the function value at that point
• Limits have applications in defining derivatives, integrals, and analyzing asymptotic behavior of functions

Definition and Intuition

• A limit is the value that a function approaches as the input gets closer to a specific value
• Intuition behind limits is to understand the behavior of a function near a point without necessarily knowing the function's value at that point
• Limits can be thought of as the "target value" that a function gets arbitrarily close to as the input approaches a certain value
• Limits are denoted using the notation $\lim_{x \to a} f(x) = L$, which means as $x$ approaches $a$, $f(x)$ gets closer to $L$
• The function does not need to be defined at the point where the limit is being evaluated
• The limit of a function may not always exist (oscillating functions or unbounded functions)
• Limits can be one-sided (left-hand or right-hand) or two-sided (the limit from both directions)

Types of Limits

• Two-sided limits consider the behavior of the function as the input approaches the point from both directions
  • For a two-sided limit to exist, both left-hand and right-hand limits must exist and be equal
• One-sided limits (left-hand and right-hand limits) describe the behavior of the function as the input approaches the point from one direction
  • Left-hand limit: $\lim_{x \to a^-} f(x)$ considers the function's behavior as $x$ approaches $a$ from the left
  • Right-hand limit: $\lim_{x \to a^+} f(x)$ considers the function's behavior as $x$ approaches $a$ from the right
• Infinite limits occur when the function grows without bound as the input approaches a certain value
  • Vertical asymptotes can be identified using infinite limits
• Limits at infinity describe the function's behavior as the input grows without bound (tends to positive or negative infinity)
  • Horizontal asymptotes can be identified using limits at infinity

Techniques for Evaluating Limits

• Direct substitution is the simplest method, where the limit value is directly substituted into the function
  • This method works when the function is continuous at the point of interest
• Factoring can be used to simplify rational functions and cancel common factors before evaluating the limit
  • Useful when direct substitution results in an indeterminate form (0/0)
• Rationalization involves multiplying the numerator and denominator by the conjugate of the denominator to simplify the expression
  • Helps eliminate square roots or nth roots in the denominator
• Trigonometric identities can be applied to simplify trigonometric functions before evaluating the limit
• Squeeze Theorem (Sandwich Theorem) is used when the function is "squeezed" between two other functions with known limits
  • 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 $\lim_{x \to a} f(x) = L$
• L'Hôpital's Rule is used to evaluate limits of indeterminate forms (0/0 or ∞/∞) by differentiating the numerator and denominator separately

Common Limit Problems

• Evaluating limits of polynomial functions usually involves direct substitution
• Limits of rational functions may require factoring or rationalization to simplify the expression
• Limits involving absolute values can be evaluated by considering the left-hand and right-hand limits separately
• Limits of trigonometric functions often require the use of trigonometric identities or special limits (e.g., $\lim_{x \to 0} \frac{\sin x}{x} = 1$)
• Limits of exponential and logarithmic functions can be evaluated using properties of exponents and logarithms
• Limits involving indeterminate forms (0/0, ∞/∞, 0⋅∞, ∞-∞, 1^∞, ∞^0, 0^0) require the application of L'Hôpital's Rule or other techniques
• Limits at infinity and infinite limits are used to determine the asymptotic behavior of functions

Continuity and Limits

• A function is continuous at a point if the limit of the function exists at that point and equals the function value
  • Mathematically, $f(x)$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$
• For a function to be continuous at a point, it must satisfy three conditions:
  • The function is defined at the point
  • The limit of the function exists at the point
  • The limit equals the function value at the point
• Discontinuities can be classified as removable, jump, or infinite discontinuities
  • Removable discontinuities occur when the limit exists, but the function is not defined or does not match the limit at that point
  • Jump discontinuities occur when the left-hand and right-hand limits exist but are not equal
  • Infinite discontinuities occur when the limit does not exist due to the function growing without bound near the point
• Continuity on an interval requires the function to be continuous at every point within the interval
• Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and y is between f(a) and f(b), then there exists a c in [a, b] such that f(c) = y

Applications in Calculus

• Limits are fundamental in defining derivatives and integrals, two essential concepts in calculus
• The derivative of a function at a point is defined as the limit of the difference quotient as the change in input approaches zero
  • $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
• The definite integral of a function over an interval [a, b] is defined as the limit of Riemann sums as the number of subintervals approaches infinity
  • $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$
• Limits are used to analyze the asymptotic behavior of functions, which is essential in curve sketching and understanding the long-term behavior of mathematical models
• Limits help determine the convergence or divergence of sequences and series
• Limits are used in the definition of continuity, which is a crucial property for many theorems in calculus, such as the Intermediate Value Theorem and the Mean Value Theorem

Practice Problems and Tips

• Practice evaluating limits using various techniques (direct substitution, factoring, rationalization, trigonometric identities, L'Hôpital's Rule)
• Identify the type of limit (two-sided, one-sided, infinite, or at infinity) and use the appropriate method to evaluate it
• When faced with an indeterminate form, try simplifying the expression using algebra before applying L'Hôpital's Rule
• Sketch the graph of the function to gain visual intuition about the limit behavior
• Remember that the limit of a function may not always equal the function value at that point
• Pay attention to the domain of the function and any potential discontinuities
• Practice problems involving piecewise functions, absolute values, and trigonometric functions to reinforce your understanding of limits in various contexts
• Analyze the continuity of functions by checking the three conditions (function defined, limit exists, limit equals function value) at each point
• Apply limits to solve problems involving derivatives, integrals, and asymptotic behavior in calculus