All Flashcards
What is a removable discontinuity?
A point where the limit exists, but the function is undefined or has a different value.
Define continuity at a point.
A function is continuous at x=a if the limit as x approaches a exists, f(a) is defined, and the limit equals f(a).
What is a piecewise function?
A function defined by multiple sub-functions, each applying to a certain interval of the domain.
Define a limit of a function.
The value that a function approaches as the input approaches some value.
What does it mean for a function to be undefined at a point?
The function does not have a value at that specific x-value.
What is the domain of a function?
The set of all possible input values (x-values) for which the function is defined.
What is a continuous function?
A function that can be drawn without lifting your pencil.
Define left-hand limit.
The value the function approaches as x approaches a value from the left.
Define right-hand limit.
The value the function approaches as x approaches a value from the right.
What is meant by 'filling the gap' in the context of removable discontinuities?
Redefining the function's value at the point of discontinuity to be equal to the limit at that point, thus making the function continuous.
Explain the concept of removing a discontinuity.
It involves redefining the function at the point of discontinuity to equal the limit at that point, effectively 'filling the hole' and making the function continuous.
Explain how factoring helps in identifying removable discontinuities.
Factoring allows you to simplify rational functions and cancel out common factors that cause discontinuities, revealing the simplified function and the location of the hole.
Describe the conditions needed for a piecewise function to be continuous at a boundary.
The left-hand limit and the right-hand limit at the boundary must exist and be equal to each other, and the function's value at the boundary must also be equal to this limit.
Explain why a removable discontinuity is not a 'true' discontinuity.
Because the limit exists at that point, and by redefining the function at that point, we can make the function continuous.
How does the graph of a function with a removable discontinuity look?
The graph appears continuous except for a single point (a 'hole') where the function is undefined.
Why is it important to check both left-hand and right-hand limits for piecewise functions?
Because the function's behavior may be different on either side of the boundary point, and continuity requires these limits to match.
What are the key steps to determine if a rational function has a removable discontinuity?
Factor the numerator and denominator, cancel any common factors, and identify the x-values that were canceled out.
Explain how the concept of a limit is essential for understanding removable discontinuities.
The limit tells us where the function 'should' be, even if it isn't defined there. This allows us to redefine the function and remove the discontinuity.
How does visualizing a graph help in identifying discontinuities?
Graphs provide a visual representation of where a function might have breaks, jumps, or holes, making it easier to spot potential discontinuities.
Explain why setting the numerator and denominator equal to zero is important.
Setting the denominator equal to zero helps to find potential points of discontinuity, and setting the numerator equal to zero helps to find the x-intercepts of the function.
What are the differences between removable and non-removable discontinuities?
Removable: Limit exists, can be 'fixed'. Non-removable: Limit doesn't exist, cannot be 'fixed'.
Compare jump and infinite discontinuities.
Jump: Function 'jumps' from one value to another. Infinite: Function approaches infinity (vertical asymptote).
What are the differences between direct substitution and factoring when evaluating limits?
Direct Substitution: Simple, works if function is continuous. Factoring: Used when direct substitution leads to indeterminate form.
Compare the graphs of continuous and discontinuous functions.
Continuous: No breaks, holes, or jumps. Discontinuous: Has breaks, holes, jumps, or asymptotes.
What are the differences between finding limits graphically and algebraically?
Graphically: Visual estimation of where the function is heading. Algebraically: Using techniques like factoring or rationalizing to find the exact value.
Compare the conditions for continuity at a point versus over an interval.
At a point: Limit exists and equals the function's value. Over an interval: Continuous at every point in the interval.
What are the differences between a hole and a vertical asymptote on a graph?
Hole: Removable discontinuity where the limit exists. Vertical Asymptote: Non-removable discontinuity where the function approaches infinity.
Compare the methods for finding discontinuities in rational functions versus piecewise functions.
Rational Functions: Look for zeros in the denominator. Piecewise Functions: Check the boundaries between the pieces.
What are the differences between the Intermediate Value Theorem and the Extreme Value Theorem?
Intermediate Value Theorem: Guarantees a function takes on every value between any two given points if continuous. Extreme Value Theorem: Guarantees the function has a max and min if continuous.
Compare direct substitution and L'Hopital's rule for evaluating limits.
Direct Substitution: First attempt, works if not indeterminate. L'Hopital's Rule: Used for indeterminate forms like 0/0 or ∞/∞, involves derivatives.