Understanding Limits in Calculus

Limits

• Definition: A limit is the y-value that a function approaches as the x-value gets closer and closer to a certain value.
• Notation: lim (x → a) f(x) = L, where:
  • lim represents the limit
  • x → a means x approaches 'a'
  • f(x) is the function
  • L is the limit value
• Limits Do Not Care About Function's Definition: Limits only consider the value the function approaches, not whether it's defined at that specific point.
• Continuity: A function is continuous at a point if it has no gaps, holes, or jumps at that point.
• Evaluating Limits Graphically: Visualize the graph to see what y-value the function approaches as x approaches the target value.
• Evaluating Limits Using a Table of Values: Create a table where x-values approach the target value from both sides. Observe the corresponding f(x) values to see if they converge to a single value.
• Evaluating Limits Algebraically:
  • Step 1: Attempt direct substitution (plug in the target x-value).
  • Step 2: If you get an indeterminate form (e.g., 0/0, infinity/infinity), use the replacement theorem.
  • Step 3: Use algebraic manipulation to eliminate the removable discontinuity and then plug in the target value.
• Indeterminate Forms: Indeterminate forms (0/0, infinity/infinity,...) indicate that further analysis is required to determine the limit.
• Limits That Do Not Exist: Limits do not exist if the function approaches different values from the left and the right of the target value, or if the function has a vertical asymptote at the target value.
• Replacement Theorem: Two functions with the same graph except for a finite number of removable discontinuities have the same limits everywhere.
• Continuity at a Point: A function is continuous at a point if:
  • The function is defined at the point.
  • The limit as x approaches the point exists (the limit from the left equals the limit from the right).
  • The limit as x approaches the point equals the function's value at that point.
• Intermediate Value Theorem (IVT):
  • Conditions: If a function is continuous on a closed interval [a, b].
  • Conclusion: The function must achieve every y-value between f(a) and f(b) at some point within the interval.

Important Notes:

• Graphical and Table of Values Methods: These are introductory methods and not used on the AP exam.
• Algebraic Method: This is the most important method for evaluating limits and is used extensively on the AP exam.
• Practice Resources: Khan Academy, Princeton Review books.
• Discord Server: The creator provides access to a Discord server for additional support.

Continuous Functions

• A continuous function must pass through every number between two points on the graph.
• Discontinuous functions have gaps, meaning there are y values the function never reaches.

The Replacement Theorem

• The replacement theorem is used to find limits algebraically when direct substitution results in an indeterminate form (e.g., 0/0).
• Common methods for using the replacement theorem:
  • Factoring - Simplifying the function by cancelling common factors.
  • Multiplying by the Conjugate - Used when there are radicals in the limit.
  • Trigonometric Substitutions - Replacing expressions with equivalent trigonometric identities.

Important Trigonometric Limits

• Memorize these limits:
  • lim (x→0) sin(x)/x = 1
  • lim (x→0) (1 - cos(x))/x = 0

Limit Properties

• Limits can be broken down using algebraic operations:
  • Sum/Difference: lim (x→a) [f(x) ± g(x)] = lim (x→a) f(x) ± lim (x→a) g(x)
  • Multiplication: lim (x→a) [f(x) * g(x)] = lim (x→a) f(x) * lim (x→a) g(x)
  • Division: lim (x→a) [f(x) / g(x)] = lim (x→a) f(x) / lim (x→a) g(x)
  • Constant Multiple: lim (x→a) [c * f(x)] = c * lim (x→a) f(x)

Hierarchy of Speeds

• Functions grow at different rates.
• The hierarchy can be used to determine limits at infinity:
  • If the numerator grows faster than the denominator, the limit is ∞.
  • If the denominator grows faster than the numerator, the limit is 0.

Difference Quotient

• The difference quotient is a key concept in calculus that leads to the derivative.
• Two forms of the difference quotient:
  • lim (x→a) [f(x) - f(a)]/(x - a)
  • lim (Δx→0) [f(x + Δx) - f(x)]/ Δx
• Direct substitution with the difference quotient always results in an indeterminate form.
• Therefore, you must use the replacement theorem to evaluate limits with the difference quotient.

This quiz explores the concept of limits in calculus, covering definitions, notations, and methods for evaluating limits graphically, with tables, and algebraically. Test your understanding of continuity and the behaviour of functions as they approach certain values.