♾️AP Calculus AB/BC Unit 1 – Limits and Continuity

Key Concepts

• Limits describe the behavior of a function as the input approaches a certain value
• Continuity refers to a function being defined at every point within its domain without any breaks or gaps
• One-sided limits consider the function's behavior as the input approaches a value from either the left or right side
• Infinite limits occur when the output of a function grows arbitrarily large or small as the input approaches a certain value
  • Vertical asymptotes are associated with infinite limits and represent a line that the function approaches but never reaches
• Limit laws and properties enable the evaluation and simplification of complex limit expressions
• Applications of limits include analyzing the behavior of functions in real-world scenarios and solving optimization problems

Limit Definition and Notation

• The limit of a function $f(x)$ as $x$ approaches a value $a$ is denoted as $\lim_{x \to a} f(x) = L$
• This notation means that as $x$ gets closer to $a$ (but not necessarily equal to $a$), the output $f(x)$ gets arbitrarily close to $L$
• The limit does not depend on the function's value at $x = a$, but rather the behavior of the function near $a$
• Limits can be evaluated from both the left and right sides of $a$, denoted as $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$, respectively
• For a limit to exist, the left-hand and right-hand limits must be equal
• The limit of a function can exist even if the function is undefined at the point of interest

Evaluating Limits

• Direct substitution can be used to evaluate limits when the function is continuous at the point of interest
  • Simply substitute the value of $a$ into the function $f(x)$ to find the limit
• Factoring and simplifying the function can help evaluate limits when direct substitution results in an indeterminate form (e.g., $\frac{0}{0}$ or $\frac{\infty}{\infty}$)
• L'Hôpital's Rule can be applied to evaluate limits of indeterminate forms involving quotients of functions
  • The rule states that $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the limit on the right-hand side exists
• Squeeze Theorem can be used to evaluate limits by comparing the function with two other functions that have known limits
• Trigonometric identities and special limits (e.g., $\lim_{x \to 0} \frac{\sin x}{x} = 1$) can simplify the evaluation of limits involving trigonometric functions

One-Sided Limits

• One-sided limits consider the behavior of a function as the input approaches a value from either the left or right side
• The left-hand limit of a function $f(x)$ as $x$ approaches $a$ is denoted as $\lim_{x \to a^-} f(x)$
  • This limit considers the function's behavior as $x$ approaches $a$ from values less than $a$
• The right-hand limit of a function $f(x)$ as $x$ approaches $a$ is denoted as $\lim_{x \to a^+} f(x)$
  • This limit considers the function's behavior as $x$ approaches $a$ from values greater than $a$
• For a limit to exist, both the left-hand and right-hand limits must be equal
• One-sided limits are particularly useful when analyzing piecewise-defined functions or functions with jump discontinuities

Infinite Limits and Asymptotes

• Infinite limits occur when the output of a function grows arbitrarily large or small as the input approaches a certain value
• Vertical asymptotes are associated with infinite limits and represent a line that the function approaches but never reaches
  • The vertical asymptote occurs at the $x$-value where the denominator of a rational function equals zero
• Horizontal asymptotes describe the behavior of a function as the input grows arbitrarily large or small
  • For rational functions, the horizontal asymptote is determined by comparing the degrees of the numerator and denominator polynomials
• Oblique (or slant) asymptotes occur in rational functions when the degree of the numerator is one less than the degree of the denominator
• Limits at infinity can be evaluated using techniques such as dividing by the highest power of $x$ in the numerator and denominator

Continuity and Types of Discontinuities

• A function is continuous at a point $a$ if the following conditions are met:
  1. The function is defined at $a$
  2. The limit of the function as $x$ approaches $a$ exists
  3. The limit of the function as $x$ approaches $a$ is equal to the function value at $a$
• Discontinuities occur when at least one of the continuity conditions is not satisfied
• Removable discontinuities (or point discontinuities) occur when the function is undefined at a point, but the limit exists
  • These discontinuities can be "removed" by redefining the function value at that point
• Jump discontinuities occur when the left-hand and right-hand limits at a point exist but are not equal
• Infinite discontinuities occur when the limit of the function as $x$ approaches a point is infinite (vertical asymptote)
• Continuity on an interval requires the function to be continuous at every point within that interval

Limit Laws and Properties

• Limit laws allow for the evaluation and simplification of complex limit expressions
• Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
• Difference Rule: $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
• Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
• Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$
• Power Rule: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$, where $n$ is a positive integer
• Constant Multiple Rule: $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$, where $c$ is a constant
• Squeeze Theorem: If $f(x) \leq g(x) \leq h(x)$ for all $x$ near $a$ (except possibly at $a$), and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$

Applications and Problem-Solving

• Limits can be used to analyze the behavior of functions in real-world scenarios, such as determining the velocity and acceleration of an object at a specific time
• Optimization problems often involve finding the maximum or minimum value of a function within given constraints
  • Limits can help identify the function's behavior near critical points and at the boundaries of the constraint intervals
• Tangent line approximations use the concept of limits to estimate the value of a function near a point
  • The slope of the tangent line is determined by the limit of the difference quotient: $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
• Limits are fundamental in defining the derivative and integral of a function in calculus
  • The derivative of a function $f(x)$ at a point $a$ is defined as: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
• Limits can be used to determine the area under a curve by approximating the region with rectangles and taking the limit as the width of the rectangles approaches zero (Riemann sums)