All Flashcards
How to find and remove a discontinuity in ?
Factor the numerator: . Cancel common factors: . Redefine .
How to ensure continuity of ?
Set at . Solve for : , so .
How to determine if has a removable discontinuity?
Factor: . Cancel: . Yes, at .
How to redefine to be continuous?
Factor: . Cancel: . Define .
How to find the value of 'a' to make continuous?
Set at . Solve for : , so .
Given , how do you find and remove the discontinuity?
Factor the numerator: . Cancel the common factor: . Evaluate at x = 1: .
How do you ensure the function is continuous?
Set the two pieces equal at x = 2: . Solve for c: , so , and .
How do you find the limit of as x approaches -5?
Factor the numerator: . Cancel the common factor: . Evaluate at x = -5: .
How do you find the value of 'k' that makes continuous?
Factor the numerator: . Cancel the common factor: . Evaluate at x = 3: . Set k = 6.
How do you determine if the function has a removable discontinuity and remove it?
Factor the numerator: . Cancel the common factor: . Yes, it has a removable discontinuity at x = -2. Evaluate at x = -2: .
How do you test continuity for a piecewise function at x=a?
What is the general form of a rational function where removable discontinuities often occur?
, where p(x) and q(x) are polynomials.
How do you find the value to 'fill the gap' at a removable discontinuity?
, where 'a' is the x-value of the discontinuity.
Write the simplified form of a function with a removable discontinuity at x=c after factoring.
If , then the simplified function is for .
What is the condition for the existence of a limit at a point?
How can you express a piecewise function generally?
How do you find the x-value(s) where a rational function might have discontinuities?
Solve where
How to redefine f(x) to remove discontinuity at x=a?
What is the simplified function after removing the discontinuity?
How do you solve for constant 'k' to make piecewise function continuous?
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.