All Flashcards
What is the Sum Rule for limits?
What is the Difference Rule for limits?
What is the Constant Multiple Rule for limits?
What is the Product Rule for limits?
What is the Quotient Rule for limits?
, if lim_{x o c} g(x)
eq 0
What is the Power Rule for limits?
What is the Root Rule for limits?
Formula to find where k is a constant?
Formula to find ?
Formula to find ?
Explain the concept of direct substitution for evaluating limits.
First attempt to evaluate a limit by plugging in the value that x approaches. If it results in a defined value, that is the limit.
What should you do if direct substitution results in an indeterminate form?
Use algebraic manipulation (factoring, rationalizing, etc.) or other techniques to simplify the expression before evaluating the limit.
Explain how limit laws help in evaluating complex limits.
Limit laws allow you to break down complex limits into simpler parts by dealing with sums, differences, products, quotients, and powers separately.
Explain the importance of checking the denominator when using the Quotient Rule.
The Quotient Rule can only be applied if the limit of the denominator is not zero. Otherwise, the limit does not exist or requires further analysis.
What happens to the limit when there is no 'x' in the function?
If the function is a constant, the limit is simply that constant value, regardless of what x approaches.
Explain the concept of one-sided limits.
One-sided limits examine the behavior of a function as x approaches a value from the left (x -> c-) or from the right (x -> c+).
How do one-sided limits relate to the existence of a two-sided limit?
For a two-sided limit to exist at a point, both the left-hand limit and the right-hand limit must exist and be equal at that point.
What is the first step in evaluating any limit?
Attempt direct substitution. If it yields a defined value, the limit is found. If it yields an indeterminate form, further techniques are required.
When can you apply the limit laws?
Limit laws can be applied when the individual limits of the functions involved exist. They help simplify complex expressions into manageable components.
Explain the relationship between limits and continuity.
For a function to be continuous at a point, the limit must exist at that point, the function must be defined at that point, and the limit must equal the function's value at that point.
What is a limit?
The value that a function approaches as the input approaches some value.
What is direct substitution in the context of limits?
Evaluating a limit by plugging in the value that x is approaching into the function.
What is an indeterminate form?
An expression whose value cannot be determined, such as 0/0.
Define the Sum Rule for limits.
The limit of a sum is the sum of the limits:
Define the Difference Rule for limits.
The limit of a difference is the difference of the limits:
Define the Constant Multiple Rule for limits.
The limit of a constant times a function is the constant times the limit of the function:
Define the Product Rule for limits.
The limit of a product is the product of the limits:
Define the Quotient Rule for limits.
The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero: , if lim_{x o c} g(x)
eq 0
Define the Power Rule for limits.
The limit of a function raised to a power is the limit of the function raised to that power:
Define the Root Rule for limits.
The limit of a root of a function is the root of the limit of the function: