Podcast
Questions and Answers
What is the limit of the function $ rac{x^2 - 9}{x - 3}$ as $x$ approaches 3?
What is the limit of the function $ rac{x^2 - 9}{x - 3}$ as $x$ approaches 3?
- 3
- Indeterminate
- 0
- 6 (correct)
Using substitution, what is the value of the limit $ rac{x + 2}{x o 4}$?
Using substitution, what is the value of the limit $ rac{x + 2}{x o 4}$?
- 6 (correct)
- 5
- 4
- 8
Which of the following statements about the limit of $ rac{x^2 - 4}{x - 2}$ as $x$ approaches 2 is correct?
Which of the following statements about the limit of $ rac{x^2 - 4}{x - 2}$ as $x$ approaches 2 is correct?
- It equals 0.
- It equals 2.
- It must be evaluated by factoring. (correct)
- It does not exist.
What is the correct approach to finding the limit of the function $f(x) = 2x + 1$ as $x$ approaches 2?
What is the correct approach to finding the limit of the function $f(x) = 2x + 1$ as $x$ approaches 2?
If the limit of $ rac{x^2 - 9}{x - 3}$ is indeterminate, what must be done to resolve it?
If the limit of $ rac{x^2 - 9}{x - 3}$ is indeterminate, what must be done to resolve it?
What does the limit of a function represent as the variable approaches a particular value?
What does the limit of a function represent as the variable approaches a particular value?
In the expression lim f(x) = L as x approaches c, what does L represent?
In the expression lim f(x) = L as x approaches c, what does L represent?
What can be concluded if for lim (3x - 2) as x approaches 2, the estimated values are 3.7, 3.97, 3.997, ?, 4.003, 4.03?
What can be concluded if for lim (3x - 2) as x approaches 2, the estimated values are 3.7, 3.97, 3.997, ?, 4.003, 4.03?
If you substitute x = 3 in the limit lim (2/(3-x)) as x approaches 3, what is the result?
If you substitute x = 3 in the limit lim (2/(3-x)) as x approaches 3, what is the result?
Which statement is true about the behavior of functions as x approaches a constant c?
Which statement is true about the behavior of functions as x approaches a constant c?
Flashcards
Limit of a Function
Limit of a Function
The value a function approaches as its input (x) gets infinitely close to a specific value (c), but never actually reaches it.
Limit Existence
Limit Existence
A limit of a function exists when the function approaches the same value from both the left and right sides of the input value.
Estimating Limit Numerically
Estimating Limit Numerically
A process where we evaluate a function for values closer and closer to a specific value (c), but never at that value itself, to see what value the function approaches.
Substitution Method
Substitution Method
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Limit Does Not Exist
Limit Does Not Exist
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Indeterminate form
Indeterminate form
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Factoring Technique
Factoring Technique
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Table of values (in limits)
Table of values (in limits)
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Study Notes
Basic Calculus - Limits
- Limits are fundamental to calculus, considered the backbone of the field, often referred to as the Mathematics of Change.
- Studying limits is crucial for understanding change in detail.
- Evaluating a limit forms the basis for calculating derivatives and integrals.
Defining Limits
- Limits focus on the behavior of a function as its variable approaches a specific value (a constant).
- The variable's value can only get very close to the constant but never equal the constant.
- The limit clarifies the function's behavior near the constant.
Definition of Limit
- If a function, f(x), gets arbitrarily close to a unique number L as x approaches c from either direction, then the limit of f(x) as x approaches c is L.
Notation
- The limit of f(x) as x approaches c is written as lim f(x) = L x→c
Examples of Limits
- Calculating limits numerically (using values close to c)
- Calculating limits algebraically (substitution and factoring)
- Example: Limit of (3x² as x approaches 4 = 48
- Example: Limit of 2x / (x² + 1) as x approaches 0 = 0
- Example of a limit that does not exist involves division by zero
Classroom Rules
- Raise your hand to answer questions.
- Do not use cell phones during class.
- Respect your fellow classmates.
Examples of Calculating Limits
- Example 1: numerically estimating lim(3x – 2) as x approaches 2 = 4
- Example 2: substitution method to find that lim(2/(3 – x) as x approaches 3 does not exist (undefined).
- Example 3: factoring and substitution to show lim ((x² – 9)/(x – 3)) as x approaches 3 = 6
- Example 4: numerically and algebraically determining lim(x + 2) as x approaches 4 = 6
Additional Problems
- Problem 1: Find the limit of (2x + 1) as x approaches 2 via substitution = 5
- Problem 2: Find the limit of (2x – 6) as x approaches 4 via substitution = 2
- Problem 3: Find the limit of ((x² – 4)/(x – 2)) as x approaches 2 (use factoring) = 4
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