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AP Calculus AB: 2.1.2 Finding Limits Graphically

Mathematics9 CardsCreated 8 months ago

Finding limits graphically involves observing the y-value a function approaches as x gets close to a specific point from both sides. If the left and right sides approach the same value, the limit exists; if they differ, the limit does not exist.

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Finding Limits Graphically

  • In algebra, you consider how a function is defined at specific points. In calculus, you can consider the value that a function approaches around a specific point.

  • The limit is the range value that a function is tending towards as you get closer to a particular domain value. If a function approaches the same value from both directions, then that value is the limit of the function at that point. If the function approaches different values, then the limit is undefined.

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Key Terms

Term
Definition

Finding Limits Graphically

  • In algebra, you consider how a function is defined at specific points. In calculus, you can consider the value that a function approaches a...

note

  • The graph of a functionis a visual way to represent the connection between the domain and the range.

  • In algebra, functions a...

Does h(x) have a limit as x approaches 1?

No, the limit doesn’t exist.

Does f(x) have a limit as x approaches 2?

Yes, the limit exists.

Does g(x) have a limit as x approaches −1?

Yes, the limit exists.

Consider the piecewise function
f(x)={1, x>0
−1, x<0
What is the limit of f(x) as x approaches -3?

lim f(x)x→−3=−1

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AP Calculus AB: 2.1.2 Finding Limits Graphically
TermDefinition

Finding Limits Graphically

  • In algebra, you consider how a function is defined at specific points. In calculus, you can consider the value that a function approaches around a specific point.

  • The limit is the range value that a function is tending towards as you get closer to a particular domain value. If a function approaches the same value from both directions, then that value is the limit of the function at that point. If the function approaches different values, then the limit is undefined.

note

  • The graph of a functionis a visual way to represent the connection between the domain and the range.

  • In algebra, functions are graphed by plotting specific points. In effect, you limit yourself to a single value of the function.

  • However, in calculus you will consider how the function behaves around a given point instead of at that point.

  • Consider running your fingers towards a certain x-value on the curve of a function, getting closer and closer to that value, but never actually touching it.

  • One possibility is that your fingers will approach different values. Notice that the calculus view of this function illustrates that something strange is going on at the point in question.

  • Another possibility is that your fingers will approach the same value. The calculus view of this function illustrates that although the function might behave strangely here, it is predictable immediately around the point.

  • The y-value that the function approaches is called the limit of that function. If no specific value is approached, the limit is said not to exist.

Does h(x) have a limit as x approaches 1?

No, the limit doesn’t exist.

Does f(x) have a limit as x approaches 2?

Yes, the limit exists.

Does g(x) have a limit as x approaches −1?

Yes, the limit exists.

Consider the piecewise function
f(x)={1, x>0
−1, x<0
What is the limit of f(x) as x approaches -3?

lim f(x)x→−3=−1

Consider the piecewise function
f(x)={1, x>0
−1, x<0
What is the limit of f(x) as x approaches 0?

The limit does not exist.

In order to answer the question “What is the limit of the function
f (x) as x approaches 3?”, you need to know:

The behavior of f (x) near x = 3, but not at x = 3.

What is the limit of g (x) as x approaches 1?

limg(x)x→1=−1