Limit
Definition
The limit of a function is the value that the function approaches as the input approaches a certain value or infinity. It represents the behavior of the function near a specific point.
Analogy
Imagine you are trying to reach a destination, but there's construction on the road. The limit is like your GPS telling you how close you can get to your destination before having to take a detour.
Related terms
Continuity: A function is continuous if its graph has no breaks, holes, or jumps. It means that the limit of the function exists at every point within its domain.
Derivative: The derivative of a function represents its rate of change at any given point. It is closely related to limits and helps analyze functions in calculus.
Infinitesimal: An infinitesimal refers to an extremely small quantity that approaches zero but is not exactly zero. In calculus, limits are used to handle infinitesimals and understand their behavior.
"Limit" appears in:
Subjects (2)
Study guides (5)
AP Calculus AB/BC - 1.8 Determining Limits Using the Squeeze Theorem
AP Calculus AB/BC - 1.12 Confirming Continuity over an Interval
AP Calculus AB/BC - 1.13 Removing Discontinuities
AP Calculus AB/BC - 1.14 Connecting Infinite Limits and Vertical Asymptotes
AP Calculus AB/BC - 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
Practice Questions (20+)
- What does a limit represent in calculus?
- For the function f(x) = x + 1, what is the limit of f(x) as x approaches 4?
- What does the limit of a function represent as x approaches infinity?
- If the limit of a function as x approaches a certain value exists, does it guarantee that the function is defined at that point?
- What is the limit of the function f(x) = 2x - 3 as x approaches infinity?
- If the limit of a function as x approaches a certain value is equal to the value of the function at that point, what can be concluded?
- What does it mean if the limit of a function does not exist?
- Which statement accurately describes the behavior of a function with a limit of 0 as x approaches infinity?
- What is the limit of the function f(x) = 3x^2 - 2x + 1 as x approaches 2?
- Which of the following statements is true regarding the limit of a constant function?
- If the limit of a function as x approaches a certain value is L, and the limit as x approaches that value from the opposite side is M, what can be concluded?
- What is the limit of the function f(x) = sin(x) / x as x approaches 0?
- What is the limit of the function f(x) = 1/x as x approaches infinity?
- Consider the graph of a function f. As x approaches a certain value from both sides, the y-values converge to the same value. What can be concluded about the limit of f as x approaches that value?
- For a function f, as x approaches a certain value from both sides, the y-values diverge and do not approach the same value. What can be concluded about the limit of f as x approaches that value?
- If the y-values from the left side and the right side of a specific x-value are different on a graph, what can be concluded about the limit at that x-value?
- On a graph, the function f(x) approaches positive infinity as x approaches a certain value. What can be concluded about the limit of f as x approaches that value?
- When estimating a limit value from a graph, if the y-values from the left and right sides approach different infinities (positive and negative), what can be concluded about the limit?
- For a function f, as x approaches a certain value, the y-values oscillate between two distinct values. What can be concluded about the limit of f as x approaches that value?
- When estimating a limit value from a graph, if the y-values from the left and right sides approach different finite values, what can be concluded about the limit?
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