All Flashcards
What does a hole in the graph of a function indicate about its limit?
It indicates a removable discontinuity. The limit may exist even though the function is not defined at that point.
What does a vertical asymptote on the graph of a function tell us about its limits?
It indicates that the limit as x approaches that value is infinite or does not exist.
How can you identify a jump discontinuity from a graph?
The graph will have a sudden 'jump' in value at a particular x-value, with different one-sided limits.
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.
What are the differences between removable and infinite discontinuities?
Removable: Limit exists, can be 'fixed'. Infinite: Vertical asymptote, limit DNE.
What are the differences between one-sided and two-sided limits?
One-sided: Limit from left or right. Two-sided: Limit from both sides, must be equal for the limit to exist.
What are the differences between average and instantaneous rate of change?
Average: Slope of secant line. Instantaneous: Slope of tangent line.