Chapter 2 Flashcards
(25 cards)
What is a function?
A relationship between an independent and dependent variable
What is a piece-wise function?
A function that behaves differently based on the input (x) value
What is a one-sided limit?
Either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right
What is a left-handed limit?
When x approaches a from the left
what is a right-handed limit?
When x approaches a from the right
Can a function have multiple different limits? If so, how?
yes, a piece-wise function can. This is because a piecewise function is not continuous
When do one-sided limits occur?
When there is an issue with continuity in a function or when the two sides do not match
When do we find infinite limits?
When the values of a function can be made arbitrarily larger, approaching an asymptote
what is a vertical asymptote?
A vertical line is an asymptote if it makes the function grow in either direction of infinity (If it is a line that the function approaches, but never meets)
When a function is approaching a limit or asymptote from the left, is it increasing or decreasing?
increasing
When a function is approaching a limit or asymptote from the right, is it increasing or decreasing?
decreasing
What is the sum law for limits?
The limit of the sum is the sum of the limits
What is the difference law for limits?
The limit of a difference is the difference of the limits
What is the constant multiple law for limits?
The limit of a constant times a function is the constant times the limit of a function
What is the product law for limits ?
The limit of a product is the product of the limits
What is the quotient law for limits?
The limit of a quotient is the quotient of the limits
What is rationalization?
The process of eliminating radicals and imaginary numbers from the denominator of an algebraic function
What is a conjugate binomial and how is it useful?
A conjugate binomial is a copy of the original binomial with the opposite sign in between the two terms. We can use it in function rationalization by multiplying both the numerator and denominator by the conjugate of the denominator to eliminate square roots from the denominator.
What is a rational function?
It is the ratio of two polynomials (one polynomial divided by the other)
What is the squeeze theorem?
If two functions squeeze together at a certain point, then they have the same limit at that point, and any function trapped between them does as well
What is the precise definition of a limit?
In simple terms, the precise definition of a limit is: for each epsilon (of a certain value range), there is a delta (of a certain value range). They have a similar relationship to the inputs and outputs of a function
What does epsilon represent?
The distance between f(x) and L on both sides
What does delta represent?
The distance between x and a on both sides
What is the purpose of the precise definition of a function?
It makes your limit calculations more precise; epsilon and delta put restrictions on the values of x and f(x) as they approach a and L
