All Flashcards
Explain the concept of a limit.
A limit describes the value that a function approaches as the input approaches some value, even if the function isn't defined there.
Explain the Squeeze Theorem.
If a function is 'sandwiched' between two other functions that approach the same limit, then the function in the middle also approaches that limit.
What are the three requirements for continuity at a point?
The function must be defined at the point, the limit must exist at the point, and the limit must equal the function's value at the point.
Explain how to find horizontal asymptotes.
Compare the degrees of the numerator and denominator. If equal, the asymptote is at the ratio of leading coefficients. If the denominator's degree is larger, the asymptote is at y=0.
Explain the Intermediate Value Theorem (IVT).
If a function is continuous on a closed interval [a, b], it must take on every value between f(a) and f(b) at least once within that interval.
Define Average Rate of Change (AROC).
The slope of the secant line between two points on a function.
Define Instantaneous Rate of Change (IROC).
The slope of the tangent line at a single point on a function.
What is a limit?
The value a function approaches as the input (x-value) gets closer to a certain point.
What is a one-sided limit?
The limit as x approaches a value from either the left or the right.
What is a two-sided limit?
The limit as x approaches a value from both the left and the right. Both one-sided limits must be equal for it to exist.
Define jump discontinuity.
The function 'jumps' from one value to another.
Define removable discontinuity.
A 'hole' in the graph that can be 'filled' by redefining the function.
Define infinite discontinuity.
A vertical asymptote where the function approaches infinity or negative infinity.
What does it mean for a function to be continuous at a point?
f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a).
Define vertical asymptote.
A vertical line x = a where the function approaches infinity or negative infinity as x approaches a.
How do you evaluate limits algebraically?
- Substitute. 2. Factor. 3. Find a Common Denominator. 4. Multiply by the Conjugate.
How do you determine if a function is continuous at a point?
- Check if f(a) is defined. 2. Check if the limit as x approaches a exists. 3. Check if the limit equals f(a).
How do you find vertical asymptotes of a rational function?
Set the denominator equal to zero and solve for x. These x-values are potential vertical asymptotes.
How do you use the Squeeze Theorem to find a limit?
- Find two functions that 'sandwich' the given function. 2. Show that the limits of the outer functions are equal as x approaches a certain value. 3. Conclude that the limit of the inner function is the same.
How do you apply the Intermediate Value Theorem?
- Verify the function is continuous on the closed interval [a,b]. 2. Check if the desired value is between f(a) and f(b). 3. Conclude that there exists a c in [a,b] such that f(c) equals the desired value.
How do you find the limit of a rational function as x approaches infinity?
Compare the degrees of the numerator and denominator. If equal, the limit is the ratio of leading coefficients. If the denominator's degree is larger, the limit is 0.