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Limits: Understand the concept of a limit of
a function and how it relates to continuity
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Introduction to Limits
- Definition of a limit
- Importance of limits in calculus
- Examples of real-world applications of limits
The concept of limits is fundamental to calculus and is used to understand the behavior of functions as
the input values approach a certain value, we will cover the definition of limits, its importance in calculus,
and some examples of real-world applications of limits
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This value can be approached from either side (left or right) of the input value
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The symbol "→" denotes the approach of
x towards a
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For example, limits are used to determine the derivative of a function, which is the rate of
change of the function at a specific point
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Examples of Real-World Applications of Limits
Limits have many real-world applications, such as in physics, engineering, and economics
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In engineering, limits are used to design structures that can withstand extreme conditions, such as
earthquakes
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limits are a crucial concept in calculus and have many practical applications
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Understanding the Concept of a Limit
- Explaining the concept of approaching a value
- One-sided and two-sided limits
- Limits at infinity
The concept of a limit is fundamental in calculus and is used to study the behavior of functions as their
inputs approach certain values
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Approaching a Value
When we say that a function f(x) approaches a value L as x approaches a value a, we mean that the
values of f(x) get closer and closer to L as x gets closer and closer to a
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As x approaches 1, the denominator (x - 1)
approaches 0, which would make the function undefined
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One-sided and Two-sided Limits
When we talk about the limit of a function at a point, we can consider both the left-hand and right-hand
limits
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e
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e
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If the left-hand and right-hand limits are equal, then we say that the function has a limit at that point
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For example, consider the function f(x) = |x|
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Therefore, the limit of f(x) as x approaches 0
exists and is equal to 0
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As x approaches 0 from the left, g(x) approaches
negative infinity, while as x approaches 0 from the right, g(x) approaches positive infinity
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Limits at Infinity
In addition to limits at specific points, we can also consider limits as x approaches infinity or negative
infinity
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As x becomes very large (either positively or negatively),
h(x) approaches 0
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Overall, understanding the concept of a limit is essential for understanding calculus and the behavior of
functions
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3
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A limit is the value that a
function approaches as its input approaches a certain value
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Definition of Continuity
A function is said to be continuous at a point if the limit of the function at that point exists and is equal to
the value of the function at that point
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For example, consider the function f(x) = x^2
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lim x->a f(x) = f(a)

Relationship between Limits and Continuity
The concept of limits is closely related to continuity
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However, the
converse is not always true
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For example, consider the function g(x) = 1/x
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However, the function is not continuous at x = 0 because the value of the function at that
point is undefined
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There are three types of discontinuities:
removable, jump, and infinite
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This
can happen when the function is undefined at that point or when there is a point where the function is not
defined but can be defined by removing the hole
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This function is undefined at x = 2 because the
denominator is equal to 0
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Therefore, the function has a removable discontinuity at x = 2
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This creates a jump in
the graph of the function at that point
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This function has a jump discontinuity at x = 0 because the
limit of the function as x approaches 0 from the left is negative infinity, while the limit of the function as x
approaches 0 from the right is positive infinity
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For example, consider the function k(x) = 1/(x - 1)
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To identify the type of discontinuity in a function, we need to evaluate the limit of the function as x
approaches the point of discontinuity from both the left and the right
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If the limits are not equal, then the function has a jump or infinite
discontinuity
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A
function is continuous if its limit at a certain point is equal to the value of the function at that point
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4
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Solving limits algebraically involves using basic limit laws, rationalizing
techniques, and factoring techniques to evaluate limits of functions
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These
laws include the limit laws of addition, subtraction, multiplication, division, and composition
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Example: Evaluate the limit of the function f(x) = x^2 + 3x - 2 as x approaches 2
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lim [x^2 + 3x - 2] = lim [x^2] + lim [3x] - lim [2]
Using the limit law of multiplication, we can evaluate each part of the function separately
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The goal is to eliminate the radical expression by multiplying both the
numerator and denominator by a conjugate
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Example: Evaluate the limit of the function f(x) = (sq rt(x) - 2)/(x - 4) as x approaches 4
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We need to rationalize the numerator to eliminate the radical expression
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The goal is to
factor the expression and simplify it to eliminate any common factors
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Example: Evaluate the limit of the function f(x) = (x^2 - 4)/(x - 2) as x approaches 2
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We need to factor the numerator to simplify the expression
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By understanding these concepts and techniques, we
can evaluate limits of functions and better understand the behavior of a function as it approaches a
particular value
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Solving Limits Algebraically
- Basic limit laws
- Rationalizing techniques
- Factoring techniques
In calculus, limits are an essential concept that helps us understand the behavior of a function as it
approaches a particular value
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Basic Limit Laws:
Basic limit laws are the fundamental rules that allow us to evaluate limits of functions algebraically
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Let's take a
look at each of these laws with an example
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Solution: Using the limit law of addition, we can break down the function into simpler parts
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lim [x^2] = (lim [x])^2 = 2^2 = 4
lim [3x] = 3(lim [x]) = 3(2) = 6
lim [2] = 2
Substituting these values back into the original equation, we get:
lim [x^2 + 3x - 2] = 4 + 6 - 2 = 8

Rationalizing Techniques:
Rationalizing techniques are used when we have a limit that involves a radical expression in the
numerator or denominator
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Let's see an example
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Solution: We can't directly substitute 4 into the function since it would result in division by
zero
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Multiplying both the numerator and denominator by the conjugate of the numerator, we get:
lim [(sq rt(x) - 2)/(x - 4)] = lim [(sq rt(x) - 2)/(x - 4)] * [(sq rt(x) + 2)/(sq rt(x) + 2)]
Simplifying the numerator using the difference of squares formula, we get:
lim [(sq rt(x) - 2)/(x - 4)] * [(sq rt(x) + 2)/(sq rt(x) + 2)] = lim [(x - 4)/(x - 4)(sq rt(x) +
2)]
Canceling out the common factor, we get:
lim [(x - 4)/(x - 4)(sq rt(x) + 2)] = lim [1/(sq rt(x) + 2)] = 1/4

Factoring Techniques:
Factoring techniques are used when we have a limit that involves a polynomial expression
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Let's see an example
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Solution: We can't directly substitute 2 into the function since it would result in division by
zero
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Using the difference of squares formula, we can factor the numerator as:
lim [(x^2 - 4)/(x - 2)] = lim [(x + 2)(x - 2)/(x - 2)]
Canceling out the common factor, we get:
lim [(x + 2)(x - 2)/(x - 2)] = lim (x + 2) = 4

In conclusion, solving limits algebraically involves using basic limit laws, rationalizing techniques, and
factoring techniques to evaluate limits of functions
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