All Flashcards
What is a limit?
The value a function approaches as the input approaches a specific point.
Define a left-hand limit.
The value a function approaches as x approaches a value from the left.
Define a right-hand limit.
The value a function approaches as x approaches a value from the right.
What does DNE mean in the context of limits?
Does Not Exist. The limit does not approach a single, finite value.
What is a vertical asymptote?
A vertical line that a function approaches but never touches; often indicates a limit DNE.
What is a jump discontinuity?
A point where the left-hand limit and right-hand limit exist but are not equal.
What is an oscillating function?
A function that fluctuates rapidly between two values as x approaches a certain point.
What is an unbounded function?
A function whose values increase or decrease without limit as x approaches a certain point.
What is the notation for the limit of f(x) as x approaches 'a'?
What does it mean for a function to be continuous at a point?
The limit exists at that point, the function is defined at that point, and the limit equals the function value.
Explain the concept of a limit.
A limit describes the value that a function approaches as the input approaches some value. It focuses on the behavior near a point, not necessarily at the point itself.
Explain one-sided limits.
One-sided limits examine the behavior of a function as it approaches a value from either the left (left-hand limit) or the right (right-hand limit).
When does a limit not exist?
A limit does not exist if the function is unbounded, oscillates wildly, or if the left-hand limit and right-hand limit are not equal.
How can graphs be used to estimate limits?
By observing the y-value that the function approaches as x approaches a specific value. Trace the curve from both sides to see if they converge to the same y-value.
What does a jump discontinuity indicate about limits?
A jump discontinuity indicates that the left-hand limit and the right-hand limit are different, therefore the limit at that point does not exist.
What does a vertical asymptote indicate about limits?
A vertical asymptote indicates that the function is unbounded as x approaches a certain value, meaning the limit does not exist at that point.
What is the relationship between continuity and limits?
For a function to be continuous at a point, the limit must exist at that point, the function must be defined at that point, and the limit must equal the function value.
Explain why scale is important when estimating limits from graphs.
The scale of a graph can distort the appearance of the function, potentially hiding important details or exaggerating certain behaviors, leading to incorrect limit estimations.
Why is it important to check one-sided limits?
Checking one-sided limits is crucial because the overall limit exists only if both the left-hand limit and the right-hand limit exist and are equal.
Explain the concept of oscillation in the context of limits.
Oscillation refers to a function fluctuating rapidly between two values as x approaches a specific point. If the oscillations become infinitely rapid, the limit does not exist.
What is the notation for the left-hand limit as x approaches a?
What is the notation for the right-hand limit as x approaches a?