1.5 Determining Limits Using Algebraic Properties, Unit 1, AP Calculus AB/BC

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.