All Flashcards
Sum Limit Law Formula
Difference Limit Law Formula
Product Limit Law Formula
Quotient Limit Law Formula
, provided
L'Hôpital's Rule Formula
If and , then
Constant Multiple Limit Law
Power Limit Law
Limit of a Constant
Limit of x
Squeeze Theorem Formula
If for all x near c, and , then .
What does the Squeeze Theorem state?
If for all x near a, and , then .
What does the Intermediate Value Theorem state?
If f is continuous on [a, b], then for any value N between f(a) and f(b), there exists a c in (a, b) such that f(c) = N.
What does the Extreme Value Theorem state?
If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum value on that interval.
What does L'Hôpital's Rule state?
If and (or both approach infinity), and if and exist, then .
How can the Squeeze Theorem be used to find limits?
If you can bound a function between two other functions that have the same limit, then the function in the middle must also have the same limit.
How is the Intermediate Value Theorem used?
To show that a continuous function takes on a specific value within a given interval.
How is the Extreme Value Theorem used?
To guarantee the existence of maximum and minimum values for a continuous function on a closed interval.
When can L'Hopital's Rule be applied?
When evaluating limits that result in indeterminate forms such as 0/0 or ∞/∞.
What is the Constant Multiple Rule for Limits?
The limit of a constant times a function is the constant times the limit of the function:
What is the Sum/Difference Rule for Limits?
The limit of a sum (or difference) is the sum (or difference) of the limits:
How to find given a graph of ?
- Locate 'a' on the x-axis. 2. Follow the graph as x approaches 'a' from both sides. 3. Identify the y-value that the graph approaches. 4. That y-value is the limit.
How to find given a table of values?
- Look at x-values approaching 'a' from both sides. 2. Observe the corresponding y-values. 3. If y-values approach the same number, that's the limit.
How to evaluate if direct substitution yields 0/0?
- Try to factor and simplify the expression. 2. If simplification doesn't work, consider L'Hôpital's Rule. 3. Take the derivative of the numerator and denominator. 4. Evaluate the limit again.
How to evaluate if contains a radical?
- Multiply the numerator and denominator by the conjugate of the expression containing the radical. 2. Simplify the expression. 3. Evaluate the limit.
How to find where P and Q are polynomials?
- Divide both numerator and denominator by the highest power of x in the denominator. 2. Evaluate the limit as x approaches infinity. Terms like 1/x approach 0.
How to determine if a limit exists at a point?
- Find the limit from the left. 2. Find the limit from the right. 3. If the left-hand limit equals the right-hand limit, the limit exists and is equal to that value.
How to use the Squeeze Theorem to find a limit?
- Find two functions, f(x) and h(x), such that f(x) <= g(x) <= h(x). 2. Find the limits of f(x) and h(x) as x approaches a. 3. If both limits are equal to L, then the limit of g(x) as x approaches a is also L.
How to find the limit of a composite function ?
- Find . 2. Find . 3. If this limit exists, it is the limit of the composite function.
How to deal with piecewise functions when finding limits?
- Determine which piece of the function applies as x approaches the target value. 2. Evaluate the limit using that piece of the function. 3. If the target value is the boundary, check both left and right limits.
How to choose between algebraic manipulation and direct substitution?
- First, try direct substitution. 2. If direct substitution results in an indeterminate form, then use algebraic manipulation techniques.