Podcast
Questions and Answers
What is the study of limits necessary for?
What is the study of limits necessary for?
- Calculating the slope of a tangent line
- Simplifying algebraic expressions
- Studying change in great detail (correct)
- Approximating the area under a curve
The variable in a limit can equal the constant it approaches.
The variable in a limit can equal the constant it approaches.
False (B)
In mathematical notation, how is 'the limit of f(x) as x approaches c is L' typically written?
In mathematical notation, how is 'the limit of f(x) as x approaches c is L' typically written?
lim xc f(x) = L
Limits are the backbone of calculus, and calculus is called the ______ of Change.
Limits are the backbone of calculus, and calculus is called the ______ of Change.
In evaluating $\lim_{x \to 2} (1 + 3x)$ using a table of values, what happens to the value of $f(x)$ as $x$ gets closer to 2?
In evaluating $\lim_{x \to 2} (1 + 3x)$ using a table of values, what happens to the value of $f(x)$ as $x$ gets closer to 2?
When evaluating a limit using a table of values, it is sufficient to approach the constant from one direction only.
When evaluating a limit using a table of values, it is sufficient to approach the constant from one direction only.
If the limit of a function as $x$ approaches $c$ from the left is not equal to the limit as $x$ approaches $c$ from the right, what can be concluded about the limit?
If the limit of a function as $x$ approaches $c$ from the left is not equal to the limit as $x$ approaches $c$ from the right, what can be concluded about the limit?
Limits that only consider values on one side of $c$ are referred to as ______ limits.
Limits that only consider values on one side of $c$ are referred to as ______ limits.
According to the examples, what is the limit of $|x|$ as $x$ approaches 0?
According to the examples, what is the limit of $|x|$ as $x$ approaches 0?
The limit of a function as $x$ approaches a value can always be determined by simply substituting the value into the function.
The limit of a function as $x$ approaches a value can always be determined by simply substituting the value into the function.
When is a limit said to 'not exist'?
When is a limit said to 'not exist'?
When evaluating $\lim_{x \to 4} f(x)$ where $f(x)$ approaches 5 from the left and 3 from the right, we say the limit ______.
When evaluating $\lim_{x \to 4} f(x)$ where $f(x)$ approaches 5 from the left and 3 from the right, we say the limit ______.
Using the graph of a function, what is a key indicator that the limit might not exist at a particular point?
Using the graph of a function, what is a key indicator that the limit might not exist at a particular point?
If a function is continuous at a point, the limit as $x$ approaches that point always exists and is equal to the function's value at that point.
If a function is continuous at a point, the limit as $x$ approaches that point always exists and is equal to the function's value at that point.
If $\lim_{x \to c^-} f(x) = L_1$ and $\lim_{x \to c^+} f(x) = L_2$, what must be true for $\lim_{x \to c} f(x)$ to exist?
If $\lim_{x \to c^-} f(x) = L_1$ and $\lim_{x \to c^+} f(x) = L_2$, what must be true for $\lim_{x \to c} f(x)$ to exist?
The limit laws allow us to evaluate limits without using a ______ or ______
The limit laws allow us to evaluate limits without using a ______ or ______
What does the Constant Multiple Theorem state about the limit of $k \cdot f(x)$ as $x$ approaches $c$?
What does the Constant Multiple Theorem state about the limit of $k \cdot f(x)$ as $x$ approaches $c$?
The Addition Theorem for limits applies only to the sum of two functions and not to their difference.
The Addition Theorem for limits applies only to the sum of two functions and not to their difference.
If $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, according to the Multiplication Theorem, what is $\lim_{x \to c} [f(x) \cdot g(x)]$?
If $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, according to the Multiplication Theorem, what is $\lim_{x \to c} [f(x) \cdot g(x)]$?
According to the ______ Theorem, the limit of a quotient of functions is equal to the quotient of the limits of the individual functions, provided the denominator limit is not equal to 0.
According to the ______ Theorem, the limit of a quotient of functions is equal to the quotient of the limits of the individual functions, provided the denominator limit is not equal to 0.
According to the power rule, if $\lim_{x \to c} f(x) = L$, then what is $\lim_{x \to c} (f(x))^p$ equal to?
According to the power rule, if $\lim_{x \to c} f(x) = L$, then what is $\lim_{x \to c} (f(x))^p$ equal to?
The Radical/Root Theorem applies only to square roots and not to other types of roots.
The Radical/Root Theorem applies only to square roots and not to other types of roots.
When applying the Radical/Root Theorem, what condition must be met if $n$ is even?
When applying the Radical/Root Theorem, what condition must be met if $n$ is even?
According to limit laws, $ \lim_{x \to c} k = $ ______, where k is any constant.
According to limit laws, $ \lim_{x \to c} k = $ ______, where k is any constant.
Using limit theorems, evaluate $\lim_{x \to 1} (2x + 1)$.
Using limit theorems, evaluate $\lim_{x \to 1} (2x + 1)$.
When evaluating the limit of a rational function, if the limit of the denominator is zero, the limit of the rational function always does not exist.
When evaluating the limit of a rational function, if the limit of the denominator is zero, the limit of the rational function always does not exist.
What is $\lim_{x \to 1} \sqrt{x}$?
What is $\lim_{x \to 1} \sqrt{x}$?
If $\lim_{x \to c} f(x) = 4$ then $\lim_{x \to c} (f(x))^3$ = ______.
If $\lim_{x \to c} f(x) = 4$ then $\lim_{x \to c} (f(x))^3$ = ______.
What is the first step in evaluating $\lim_{x \to 2} \frac{x}{x-1}$?
What is the first step in evaluating $\lim_{x \to 2} \frac{x}{x-1}$?
Match the limit theorems with their descriptions:
Match the limit theorems with their descriptions:
Which of the following is a direct application of the limit of a constant rule?
Which of the following is a direct application of the limit of a constant rule?
If $\lim_{x \to c} f(x)$ exists, then $f(x)$ must be defined at $x = c$.
If $\lim_{x \to c} f(x)$ exists, then $f(x)$ must be defined at $x = c$.
What theorem justifies splitting $\lim_{x \to a} (f(x) + g(x))$ into $\lim_{x \to a} f(x) + \lim_{x \to a} g(x)$?
What theorem justifies splitting $\lim_{x \to a} (f(x) + g(x))$ into $\lim_{x \to a} f(x) + \lim_{x \to a} g(x)$?
To directly apply the Division Theorem for limits, the limit of the ______ must not be zero.
To directly apply the Division Theorem for limits, the limit of the ______ must not be zero.
Given $\lim_{x \to 2} f(x) = 5$ and $\lim_{x \to 2} g(x) = -3$, find $\lim_{x \to 2} (2f(x) + g(x))$.
Given $\lim_{x \to 2} f(x) = 5$ and $\lim_{x \to 2} g(x) = -3$, find $\lim_{x \to 2} (2f(x) + g(x))$.
The limit of a function as x approaches infinity can always be found by substituting infinity into the function.
The limit of a function as x approaches infinity can always be found by substituting infinity into the function.
What is the value of $\lim_{x \to 7} x$?
What is the value of $\lim_{x \to 7} x$?
By the constant multiple theorem,$\lim_{x \to c} -11 * f(x) = -11 * \lim_{x \to c} f(x) = -11 * L$ if $\lim_{x \to c} f(x) = $ ______.
By the constant multiple theorem,$\lim_{x \to c} -11 * f(x) = -11 * \lim_{x \to c} f(x) = -11 * L$ if $\lim_{x \to c} f(x) = $ ______.
Given $\lim_{x \to c} f(x)= 4$ and $\lim_{x \to c} g(x) = -5$, calculate $\lim_{x \to c} f(x) + g(x)$.
Given $\lim_{x \to c} f(x)= 4$ and $\lim_{x \to c} g(x) = -5$, calculate $\lim_{x \to c} f(x) + g(x)$.
Flashcards
What are Limits?
What are Limits?
The backbone of calculus used to study change in detail.
What is a Limit of a Function?
What is a Limit of a Function?
A function's behavior as its variable gets close to a value.
What is L in Limit Notation?
What is L in Limit Notation?
The unique real value that f(x) approaches as x approaches c.
What is the Limit?
What is the Limit?
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What happens as x gets closer?
What happens as x gets closer?
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When Does a Limit Exist?
When Does a Limit Exist?
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What does DNE mean?
What does DNE mean?
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What are One-Sided Limits?
What are One-Sided Limits?
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What do Limit Laws enable?
What do Limit Laws enable?
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What is the limit of a constant?
What is the limit of a constant?
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Limit of a Linear Function
Limit of a Linear Function
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Constant Multiple Theorem
Constant Multiple Theorem
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Addition Theorem
Addition Theorem
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Multiplication Theorem
Multiplication Theorem
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Division Theorem
Division Theorem
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Power Limit
Power Limit
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Radical/Root Theorem
Radical/Root Theorem
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What is Application of Limit Theorems?
What is Application of Limit Theorems?
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rational functions
rational functions
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Study Notes
- The slides contain study notes on limits and limit theorems in basic calculus
- This note covers illustrating limits, distinguishing between limits, the theorems themselves, and applying the theorems to algebraic functions
Introduction
- Limits are the foundation of calculus, often called "Mathematics of Change"
- Studying limits very important for understanding change in detail
- Evaluating a particular limit is essential for the formulation of derivatives and integrals
Limits of Functions
- Functions of a single variable x are analyzed as x approaches a particular value c
- The variable x can get very close to c but cannot equal c
- The limit describes what happens to the function near that constant
Limit of a Function Using Table of Values and Graph
- Consider a function f of a single variable x
- Consider a constant c, which the variable x will approach
- c may or may not be in the domain of f
- The limit L is the unique real value that f(x) will approach as x approaches c
- In symbols, this process is written as lim (x→c) f(x) = L
- This reads as "The limit of f(x) as x approaches c is L"
- Evaluating tables helps track the effect that the approach of x toward the constant has on f(x)
Example:
- Consider lim (x→2) (1+3x)
- f(x) = (1+3x), and the constant c is 2
- As x approaches 2 from the left (values less than 2), f(x) approaches 7
- As x approaches 2 from the right (values greater than 2), f(x) also approaches 7
- Regardless of the set of values or the direction of approach, as x gets closer to 2, f(x) gets closer to 7
- This is written symbolically as lim (x→2) (1+3x) = 7
Example 1:
- Investigate lim (x→-1) (x² + 1) using tables of values
- Here, c = -1 and f(x) = x² + 1
- Approaching -1 from the left, as x gets closer to -1, f(x) approaches 2
- Approaching -1 from the right, as x gets closer to -1, f(x) approaches 2
- This suggests lim (x→-1) (x² + 1) = 2
Example 2:
- Investigate lim (x→0) |x| through a table of values
- Approaching 0 from both the left and the right, |x| approaches 0
- Thus, lim (x→0) |x| = 0
- f(x) = |x| in this example
Example 3:
- Investigate lim (x→1) (x²-5x+4 / x-1) by constructing tables of values
- Approaching 1 from the left and the right, f(x) approaches -3
- So, lim (x→1) (x²-5x+4 / x-1) = -3
Example 4:
- Investigate the limit through a table of values if f(x) = x+1 if x<4, and (x-4)²+3 if x ≥ 4
- x approaches 5 from the left while it approaches 3 from the right
- The values that f(x) approaches are not equal, and the limit does not exist (DNE)
One-Sided Limits
- For a limit L to exist, the limits from the left and right must both exist and be equal to L
- lim(x→c) f(x) DNE whenever lim(x→c⁻) f(x) ≠ lim(x→c⁺) f(x)
- Limits are referred to as one-sided limits
Limit Theorems
- Limit laws can be used to directly evaluate limits without needing a table or graph
- Let c be a constant, and f and g be functions that may or may not have c in their domains
Limit of a Constant
- The limit of a constant is the constant itself
- If k is any constant, then lim (x→c) k = k
Limit of a Linear Function
- The limit of x as x approaches c is equal to c
- This can be thought of as the substitution law because x is simply substituted by c
Constant Multiple Theorem
- The limit of a multiple of a function is simply that multiple of the limit of the function
- lim (x→c) k * f(x) = k * lim (x→c) f(x) = k * L
The Addition Theorem
- The limit of a sum of functions is the sum of the limits of the individual functions
- This includes subtraction within its laws
- In symbols, lim (x→c) (f(x) + g(x)) = lim (x→c) f(x) + lim (x→c) g(x) = L + M
- lim (x→c) (f(x) - g(x)) = lim (x→c) f(x) - lim (x→c) g(x) = L - M
Multiplication Theorem
- Similar to the addition theorem, but with multiplication replacing addition
- The limit of a product of functions equals the product of their limits
- lim (x→c) (f(x) * g(x)) = lim (x→c) f(x) * lim (x→c) g(x) = L * M
Division Theorem
- The limit of a quotient of function is equal to the quotient of the limits
- The denominator's limit is not equal to zero
- In symbols: lim (x→c) f(x)/g(x) = lim (x→c) f(x) / lim (x→c) g(x) = L/M, provided M≠ 0
The Power Limit
- The limit of an integer power p of a function is just that power of the limit of a function
- lim (x→c) (f(x))^p = (lim (x→c) f(x))^p = L^p
The Radical/Root Theorem
- If n is a positive integer, the limit of the nth root of a function is just the nth root of the limit
- Provided that nth root of the limit is a real number
- If n is even, the limit of the function must be positive
Application of Limit Theorems
- Limit theorems can evaluate algebraic functions
Limits of Polynomial Functions
- Polynomial functions can be evaluated using limit theorems
Limits of Rational Functions
- Limit theorems are applied when evaluating rational functions
- Theorem 1, the Division Rule, must be recalled
Limits of Radical Functions
- Radical functions are evaluated using limit theorems
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