AP Calculus BC Flashcards: Defining Continuity at a Point
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
If lim_{x→5} g(x) = 10 but g(5) is undefined, is the function g continuous at x=5? Justify your answer.
No, the function g is not continuous at x=5. The first condition for continuity, that g(c) must exist, is not met.
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If lim_{x→5} g(x) = 10 but g(5) is undefined, is the function g continuous at x=5? Justify your answer.
No, the function g is not continuous at x=5. The first condition for continuity, that g(c) must exist, is not met.
State the formal, three-part definition of continuity at a point.
A function f is continuous at x=c provided that f(c) exists, lim_{x→c}f(x) exists, and lim_{x→c}f(x) = f(c).
What is the final condition that must be satisfied for a function f to be continuous at x=c?
The final condition is that the limit of the function as x approaches c must equal the value of the function at c.
What are the three conditions required for a function f to be continuous at a point x=c?
The three conditions are: 1) f(c) exists, 2) the limit of f(x) as x approaches c exists, and 3) the limit of f(x) as x approaches c is equal to f(c).
A function h(x) is defined such that h(2)=4, but the lim_{x→2} h(x) does not exist. Is h(x) continuous at x=2? Justify.
No, h(x) is not continuous at x=2. The second condition for continuity, that the limit must exist at that point, is not satisfied.
What is the first step in justifying a conclusion about continuity at a point x=c?
The first step is to check if the function is defined at the point, meaning that f(c) exists.
Given that f(a) exists and lim_{x→a}f(x) exists, but f(a) ≠ lim_{x→a}f(x). Is f continuous at x=a? Justify.
No, f is not continuous at x=a because the third condition for continuity is not met; the limit as x approaches a does not equal the function's value at a.
What is the relationship between the limit and the function's value for a function to be continuous at a point?
For a function to be continuous at a point c, the limit as x approaches c must be exactly equal to the function's value at c (lim_{x→c}f(x) = f(c)).
How do you use the definition of continuity to justify your conclusion about a function's continuity at a point?
To justify a conclusion, you must explicitly check and state whether each of the three conditions of the definition of continuity is met for that specific point.
Why is it insufficient to only know that f(c) exists to determine if a function is continuous at x=c?
Knowing only that f(c) exists is not enough because continuity also requires the limit to exist at c and for that limit to be equal to the value of f(c).