1.1-1.2 Introducing Calculus: Can Change Occur at an Instant? and Defining Limits and Using Limit Notation, Unit 1, AP Calculus AB/BC

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.