Unit 1 - Calc AB Limits & Continuity

Approaching Without Reaching

Getting closer and closer to a value or point without actually reaching it.

Value the Function Approaches

This is the number that the output of a function gets closer to as the input approaches a certain value.

Limiting Value

The limiting value is the value that the outputs of a function approach as the input approaches a certain value.

Function Output Trends

Function Output Trends mean how the values of a function change or behave as the inputs approach a certain number.

Predicted Behavior

The expected output of a function near a specific input, based on observing its values nearer and nearer that input.

Closer and Closer Behavior

The idea that the function's output gets increasingly near a certain number as the input approaches a specific value.

Tending Toward a Value

Tending toward a value means that as the input changes, the output or variable moves closer and closer to a certain number, which we call the limit.

Nearby Values

Nearby values are the numbers that are close to a particular number on the number line. We look at these values to understand how a function behaves near that number.

Getting Arbitrarily Close

Getting arbitrarily close means that a value can be made as near as we want to another value by choosing inputs close enough. There is no fixed distance; we can make the difference smaller than any positive number.

Arbitrary Closeness Concept

The arbitrary closeness concept means that for any small positive number, no matter how tiny, we can find an input value close enough to a point so that the function's output is within that small distance from the limit.

Not Necessarily Reaching the Point

It means that a function can approach a value arbitrarily closely as the variable approaches a point, even if the function never equals that value or is not defined at the point itself.

Limit Expression

A math expression written using limit notation that describes the value a function approaches when the input nears a specific number.

Non-Uniqueness Implies Nonexistence

If two different limits are found when approaching a point from different sides, then the overall limit at that point does not exist.

Uniqueness of the Limit

If a limit of a function at a point exists, it is the only limit at that point; there cannot be two different limits for the same input.

Limit Symbol

The limit symbol is written as lim and is used to indicate that we are finding the limit of a function as a variable approaches a certain value.

Existence of Limit

The limit of a function at a point exists only if the values the function approaches from both sides are equal and finite as the input nears that point.

Limit Notation

Limit notation is how we write limits using symbols and expressions to describe what value a function approaches as the input approaches a certain point.

Variable Tending Expression

A variable tending expression shows the variable getting closer and closer to a specific value, often written as “𝑥 → 𝑎,” where 𝑥 is the variable and 𝑎 is the value.

Visual Interpretation of Limits

Visual interpretation of limits means understanding how function values behave near a specific point by looking at the graph of the function.

Graphical Behavior Near Point

Graphical behavior near a point shows how a function behaves on the graph when the input values are very close to that point from both sides.

Equality of Left-Hand and Right-Hand Limits

The limit at a point is the same from both directions, so the left-hand limit and the right-hand limit must be equal for the two-sided limit to exist.

Approach from Right on Graph

Approach from right on a graph refers to observing how the function behaves as the input values get close to a point from larger values on the right side.

One-Sided Limit on Graph

A one-sided limit on a graph shows the behavior of a function's values as the input approaches a specific point from only one side, either from the left or the right.

Approach from Left on Graph

Approach from left on a graph means looking at the values of a function as the input gets closer to a particular point coming from smaller values, or from the left side on the x-axis.

Right-Hand Limit

The right-hand limit of a function at a point is the value that the function approaches as the input gets closer to that point from values greater than the point. It observes the function's behavior from the right side only.

Left-Hand Limit

The left-hand limit of a function at a point is the value that the function approaches as the input gets closer to that point from values less than the point. It looks at the behavior of the function from the left side only.

Limit From Graph Shape

Limit from graph shape means estimating the limit by looking at the overall form and trend of the graph near a point. The shape suggests what value the function is getting close to, even if the function is not defined exactly at that point.

One-Sided Limits Graphically

One-sided limits graphically focus on the limit of a function as the input moves towards a point from only one side — either the left or the right. This helps understand cases where the function’s behavior is different on each side of the point.

Left-Hand and Right-Hand Limits from Graphs

Left-hand and right-hand limits from graphs are determined by observing the function values as the inputs approach a point from the left side and the right side. These limits show whether the function approaches the same value from both directions or not.

Non-Existence via Divergent One-Sided Limits

Non-existence via divergent one-sided limits happens when the limit of a function approaching a point from the left is different from the limit approaching from the right, or one of the limits does not exist. In such cases, the two-sided limit does not exist.

Comparing Tables of Values for One-Sided Limits

A table of values helps estimate one-sided limits by listing function outputs for inputs that get closer and closer to the limit point from one direction. Comparing those tables from the left side and the right side shows how function values behave near that point, which helps to understand if the two-sided limit exists or not.

Left-Hand Limit Table

A Left-Hand Limit Table lists function values for inputs approaching a point from the left side, or values less than the target point.

Pattern Recognition in Values

Observing the function values in a numerical table to see if they approach a single number as the input gets closer to the limit point.

Testing Values Closer To Limit

Testing Values Closer To Limit involves finding function outputs for inputs increasingly closer to the limit point, to better approximate the limit value.

Tabular Analysis of Limits

Tabular Analysis of Limits uses tables to organize input numbers and their function outputs to estimate what value the function approaches close to some point.

Step Size Selection

Step Size Selection is the process of choosing how close to make input values when studying limits. Smaller steps give a better estimate of the function’s limit.

Tables of 𝑥 and 𝑓(𝑥)

Organized lists showing chosen input values (𝑥) and their corresponding outputs (𝑓(𝑥)), used to help find limits by examining output trends as inputs approach a point.

Right-Hand Limit Table

A table listing input values approaching the target number from the right (larger numbers) and their corresponding function values to estimate the limit from the right side.

Successive Function Values

Successive Function Values are a series of function outputs calculated by choosing input values that approach a specific point step by step. These help us observe behavior near that point.

Limit of Quotient

The value that the ratio of two functions approaches as the input approaches a specific point, provided the denominator’s limit is not zero.

Evaluating Limits By Direct Substitution

The process of calculating the limit of a function by replacing the variable with the target value and simplifying if needed.

Nonzero Denominator Condition for Quotient Law of Limits

The requirement that the limit of the denominator function must not be zero in order to apply the Quotient Law for Limits safely.

Limit of Product

The limit of the product of two functions as the input approaches a value is equal to the product of their limits, provided both limits exist.

Limit of Cube Root

The limit of the cube root of a function is the cube root of the limit of that function because the cube root is continuous for all real numbers.

Constant Multiple Law

This law states that the limit of a constant multiplied by a function is the constant times the limit of that function, assuming the limit exists.

Function Value Equals Limit

The concept that when a function is continuous at a point, the limit as 𝑥 approaches that point is the same as the function value at that point.

Limits Involving Division

Limits involving division deal with finding limits of ratios of functions, often using the Quotient Law when the limit of the denominator is not zero.

Limit of Square Root

The limit of the square root of a function is the square root of the limit of that function, as long as the function’s limit is nonnegative and the square root function is continuous there.

Limit of Sum

The limit of the sum of two functions as 𝑥 approaches a value is equal to the sum of their individual limits. If the limits exist, you can find the limit of the sum by adding each limit separately.

Limit of 𝑓(𝑔(𝑥))

The limit of 𝑓(𝑔(𝑥)) is the value that the composite function approaches as 𝑥 gets closer to a specific point. If 𝑔(𝑥) approaches a limit 𝐿 and 𝑓 is continuous at 𝐿, then the limit of 𝑓(𝑔(𝑥)) is 𝑓(𝐿).

Root Law for Limits

Root Law for Limits states that the limit of a root of a function equals the root of the limit of that function, given the limit inside the root exists and satisfies any necessary domain restrictions.

Direct Substitution Property for Rational Functions

If a rational function is defined at a point and its denominator is not zero there, the limit of the function as the input approaches that point can be found by directly substituting the point into the function.

Direct Substitution Property for Polynomial Limits

The rule that says to find the limit of a polynomial function at a point, you can just replace the variable with the point’s value. Since polynomials are continuous, doing this substitution gives the exact limit.

Domain Considerations For Substitution

Domain Considerations For Substitution remind us that we can only substitute values into a function if those values are within the function’s domain and do not cause division by zero or other undefined situations.

Limit of Power Function

The limit of a power function can be found by applying the limit to the base and then raising the result to the given exponent, provided the power function is defined and continuous at that value.

Limit of a Polynomial

The value that a polynomial function approaches as the input approaches a certain number. Because polynomials are continuous everywhere, their limits at any point can be found by simply using that value in the function.

Limit of Difference

The limit of the difference between two functions as 𝑥 approaches a value is equal to the difference of their individual limits. If both limits exist, you subtract the two limits to get the overall limit of the difference.

Linear Combination Property of Limits

The linear combination property of limits means that the limit of 𝑎 times one function plus 𝑏 times another function equals 𝑎 times the limit of the first function plus 𝑏 times the limit of the second function, where 𝑎 and 𝑏 are constants.

Power Law for Limits

The Power Law for Limits says that the limit of a function raised to a power is equal to the limit of the function raised to that same power. In symbols, the limit of [𝑓(𝑥)]ⁿ as 𝑥 approaches a value 𝑐 is [limit of 𝑓(𝑥) as 𝑥 approaches 𝑐]ⁿ, provided the limit exists (and is nonzero if 𝑛 is negative).

Limit Existence of Rational Function at Nonzero Denominator

The limit of a rational function exists at a point if the denominator of the function is not zero at that point. When the denominator is nonzero, you can find the limit by directly substituting the value into the function because the function is continuous there.

Monomial Limit Evaluation

Monomial limit evaluation involves finding the limit of a single-term power function, such as 𝑥 raised to 𝑛. Since monomials are continuous, the limit can be found by directly substituting the value the variable approaches into the expression.

Rewriting Limit in Terms of New Variable

This technique rewrites a limit by substituting a new variable for part of the expression (often the quantity approaching zero), making the limit easier to analyze.

Algebraic Substitution

Algebraic substitution is a method where you replace a variable in an expression with another expression or value to simplify the calculation of a limit.

Limit Reformulation

Limit reformulation involves changing the form of a limit problem, often by using substitutions or algebraic manipulations, to make the limit easier to compute.

Algebraic Simplification before Evaluating Limit

This is the process of rewriting a function in an easier form by factoring or canceling common terms before finding its limit. It helps to avoid direct substitution if it leads to indeterminate forms.

Simplifying Numerator and Denominator and Then Evaluating Limit

This involves factoring or reducing the expressions in both the numerator and denominator so that the function can be simplified. After simplification, the limit is calculated, usually by direct substitution.

Reducing to Lowest Terms Before Evaluating Limit

Reducing to lowest terms means factoring and canceling common factors in a rational expression before calculating the limit. This simplifies the expression and allows the limit to be evaluated, even if the original function is not defined at that point.

Nested Radical Simplification Before Evaluating Limit

Nested radical simplification is the process of rewriting expressions that contain roots inside other roots—often by rationalizing or algebraic manipulation—before evaluating a limit. This makes the limit easier to compute.

Evaluating Limit by Cancelling Out Common Factors

This method involves factoring expressions to identify and cancel common factors that lead to indeterminate forms like 0/0. After simplification, the limit is found by direct substitution into the simplified expression.

Least Common Denominator

The least common denominator is the smallest number that can be used as a denominator for two or more fractions so they can be compared or combined. It is the least common multiple of the denominators of the fractions involved.

Combining Fractions

Combining fractions means writing two or more fractions as a single fraction. To do this, you first find the least common denominator and then write each fraction with this denominator before adding or subtracting the numerators.

Zero Limits of Root Functions

Zero limits of root functions describe what happens to expressions involving roots as the variable approaches zero. In some cases, these limits lead to indeterminate forms like 0 divided by 0, and techniques such as rationalization are used to evaluate them.

Infinity Limits of Root Functions

Infinity limits of root functions describe how expressions involving roots behave as the variable grows very large or very small. Generally, roots grow slower than powers, so analyzing their behavior at infinity helps find those limits correctly.

Rationalization Strategy for Roots

The rationalization strategy involves multiplying the numerator and denominator by a conjugate or expression to eliminate roots in the denominator or numerator. This technique transforms the limit into a simpler form that can be evaluated directly.

Numerator Rationalization and Limit Evaluation

Numerator rationalization involves multiplying the numerator and denominator by the conjugate of the numerator. This simplifies expressions when the numerator has square roots, helping to evaluate limits that would otherwise give an indeterminate form.

Radicals in Limit Expressions

Radicals in limit expressions are roots, such as square roots or cube roots, that appear in limit problems. When evaluating limits with radicals, special care is sometimes needed because direct substitution can lead to indeterminate forms like 0 divided by 0.

Denominator Rationalization and Limit Evaluation

Denominator rationalization is the process of removing roots from the denominator by multiplying both the numerator and denominator by a conjugate expression. This simplifies the limit expression and helps find the limit when substitution results in an indeterminate form.

Adding and Subtracting Rational Functions

Adding and subtracting rational functions involves combining two fractions that contain polynomials in their numerators and denominators. This process requires finding the least common denominator and rewriting each rational function with this denominator before performing the operation.

Simplifying Complex Fractions

Simplifying complex fractions involves rewriting a fraction whose numerator or denominator is itself a fraction into a simpler expression. This is usually done by finding a common denominator and reducing the expression.

Polynomial Factorization Techniques and Limits

Polynomial factorization techniques involve rewriting a polynomial as a product of simpler polynomials. This method helps simplify expressions, especially when calculating limits. By factoring, we can cancel common factors in the numerator and denominator to find the limit of a rational function more easily.

Rationalization Technique and Limit Evaluation

The rationalization technique is a method used in calculus to help find limits when expressions involve square roots. It involves multiplying the expression by a form of 1 that contains a conjugate. This helps eliminate the square root from either the numerator or denominator, making the limit easier to evaluate.

Polynomial Division and Limits of Rational Functions

Polynomial division is a method used to rewrite a rational function when the degree of the numerator is greater than or equal to the denominator. This helps in finding limits by expressing the function in a simpler form, often turning it into a polynomial plus a smaller fraction, which makes evaluating limits easier.

Common Factor Cancellation and Limit Evaluation

Common factor cancellation is a method used to simplify limit expressions. When a common factor in the numerator and denominator leads to an indeterminate form like 0 divided bu 0, factoring and canceling that factor can simplify the expression. The limit is then evaluated by substitution into the simplified expression, even if the original function is not defined at that point.

Zero Factor Property and Limits of Polynomials

The Zero Factor Property states that if a product equals zero, then at least one of its factors must be zero. In limit problems involving polynomials, this property helps us factor expressions and identify common factors in the numerator and denominator. Canceling these common factors can remove an indeterminate form like 0 divided by 0, allowing the limit to be evaluated even if the original function is not defined at that point.

Which procedure should be used to determine the limit of 𝑓(𝑥) = 3𝑥² − 5 as 𝑥 → 2?

Use direct substitution because the function is a polynomial and is continuous at 𝑥 = 2, so the limit equals the function value.

Which procedure should be used to determine the limit of (𝑥² − 4)/(𝑥 − 2) as 𝑥 → 2?

Use algebraic simplification because direct substitution gives an indeterminate form 0/0, and factoring allows the limit to be evaluated.

Which procedure should be used to determine the limit of (√(𝑥 + 2) − 2)/(𝑥 − 2) as 𝑥 → 2?

Use algebraic manipulation by multiplying by the conjugate, which removes the indeterminate form and allows evaluation of the limit.

Which procedure should be used to determine the limit of (𝑥² − 1)/(𝑥² + 1) as 𝑥 → ∞?

Use analysis of end behavior by comparing leading terms, since both numerator and denominator are polynomials of the same degree.

Which procedure should be used to determine the limit of sin(𝑥)/𝑥 as 𝑥 → 0?

Use a known trigonometric limit because direct substitution gives 0/0 and this limit is a standard foundational result.

Which procedure should be used to determine the limit of (2𝑥² + 1)/(𝑥² − 5) as 𝑥 → ∞?

Divide the numerator and denominator by 𝑥² to analyze horizontal asymptotic behavior and determine the limit at infinity.

Which procedure should be used to determine the limit of |𝑥|/𝑥 as 𝑥 → 0?

Evaluate one-sided limits because the expression behaves differently for 𝑥 > 0 and 𝑥 < 0, and the two-sided limit depends on their agreement.

Which procedure should be used to determine the limit of a piecewise function as 𝑥 → 1?

Evaluate the left-hand and right-hand limits separately using the appropriate expression on each side of 𝑥 = 1.

Which procedure should be used to determine the limit of (𝑥² + 3𝑥)/(𝑥) as 𝑥 → 0?

Simplify algebraically by canceling 𝑥 before substituting, since direct substitution leads to an indeterminate form.

Which procedure should be used to determine the limit of 𝑓(𝑥) given only its graph as 𝑥 → 𝑎?

Use graphical analysis by examining the 𝑦-values the function approaches from the left and right near 𝑥 = 𝑎.