Podcast
Questions and Answers
What is the general form of a rational function?
What is the general form of a rational function?
What is the limit of a rational function when the denominator is zero?
What is the limit of a rational function when the denominator is zero?
What is the limit of h(x) as x approaches 4, given that h(x) = x^2?
What is the limit of h(x) as x approaches 4, given that h(x) = x^2?
What is the limit of f(x) as x approaches -4, given that f(x) = x^2?
What is the limit of f(x) as x approaches -4, given that f(x) = x^2?
Signup and view all the answers
What is the limit of 2x^2 - 6x + 18 as x approaches 6?
What is the limit of 2x^2 - 6x + 18 as x approaches 6?
Signup and view all the answers
What is the limit of x^2 + 2x - 7 as x approaches 2?
What is the limit of x^2 + 2x - 7 as x approaches 2?
Signup and view all the answers
What is the limit of 2x^2 - 9x - 5 as x approaches 1?
What is the limit of 2x^2 - 9x - 5 as x approaches 1?
Signup and view all the answers
What is the limit of h(x) as x approaches a, given that h(x) is a rational function?
What is the limit of h(x) as x approaches a, given that h(x) is a rational function?
Signup and view all the answers
What is the property of limits that allows us to evaluate the limit of a sum of functions as the sum of the limits of the individual functions?
What is the property of limits that allows us to evaluate the limit of a sum of functions as the sum of the limits of the individual functions?
Signup and view all the answers
What is the purpose of the limit theorems in calculus?
What is the purpose of the limit theorems in calculus?
Signup and view all the answers
Study Notes
Definition of a Limit
- A limit is a unique real value that a function approaches as the variable x approaches a constant a.
- The limit, denoted by L, is the value that the function f(x) will approach as x approaches a.
Limit Theorems
- The limit of a function f(x) as x approaches a is denoted by lim f(x) = L.
- The limit exists if we find a single real function value upon applying the limit theorems.
- If lim f(x) = L1 and lim f(x) = L2, then L1 = L2.
One-Sided Limits
- The limit of f(x) as x approaches 2 from the right is 4.
- The limit of f(x) as x approaches 2 from the left is 4.
Informal Definition of a Limit
- The limit of a function f of a single variable x is the unique real value that f(x) approaches as x approaches a constant a.
- The limit is denoted by L, and is written as lim f(x) = L.
Limit Theorems (Examples)
- Example 1: lim (2x - 1) / (x - 1) = 2
- Example 2: lim (x^2 + x - 2) / (2x - 1) = 4
- Example 3: lim (x^2 + x - 2) / (x - 2) = 4
Existence of Limits
- The limit exists if we find a single real function value upon applying the limit theorems.
- If lim f(x) = L1 and lim f(x) = L2, then L1 = L2.
Uniqueness of Limits
- If the limit exists, it is unique.
- If lim f(x) = L1 and lim f(x) = L2, then L1 = L2.
Rational Functions
- A rational function is a function of the form f(x) = h(x) / g(x), where h(x) and g(x) are polynomial functions.
- The limit of a rational function as x approaches a is denoted by lim f(x) = L.
Radical Functions
- A radical function is a function of the form f(x) = x^n, where n is a real number.
- The limit of a radical function as x approaches a is denoted by lim f(x) = L.
Examples of Limit Theorems
- Example 1: lim x^2 = 4
- Example 2: lim x^3 = 8
- Example 3: lim x^4 = 16
Infinite Limits
- The limit of a function as x approaches a is infinite if the function values increase or decrease without bound as x approaches a.
- The limit of a function as x approaches a is denoted by lim f(x) = ∞ or lim f(x) = -∞.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Test your understanding of limits, limit theorems, and one-sided limits in calculus. Learn how to approach limits and apply limit theorems to find a single real function value.
