Chapter 2 Flashcards
Velocity
As two points get closer and closer, the velocity gets more accurate
Secant Line
A line that intersects a function in two positions
Tangent Line
A line that touches a function at one point
Limit of a Function
lim f(x) = L x -> a
The limit of f(x) as x approaches a, equals L
Cases Where Limits are Useful
- When there is a hole in the function
- When a limit is undefined (0 in denominator)
- When you have infinity/infinity
Issues When Using Limits
- Heavy Side Function - left and right side are mismatched
- Vertical Asymptope on a graph
- Functions that are ‘wild’
|—> f(x) = sin(1/x)
DNE
- When we cannot calculate a limit, it does not exist
Law of Limits: Sum Law
The limit of a sum is the sum of the limits.
lim[f(x) + g(x)] = lim f(x) + lim g(x) as x –> a
Law of Limits: Difference Law
The limit of a difference is the difference of the limits.
lim[f(x) - g(x)] = lim f(x) - lim g(x) as x –> a
Law of Limits: Constant Multiple Law
The limit of a constant times a function is the constant times the limit of the function.
lim[cf(x)] = c * lim f(x)
Law of Limits: Product Law
The limit of a product is the product of the limits.
lim[f(x)g(x)] = lim f(x) * lim g(x) as x –> a
Law of Limits: Quotient Law
The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0)
lim [f(x)/g(x) = lim f(x) / lim g(x) as x –> a
Limits: Exponents and Roots
Limits can move through exponents and roots
Strategy For Calculating Limits: 1. Plugging In
Plug in numbers that are close on the left and right.
“Guess and check”
Always plug in ‘a’ for ‘x’ values and see if it works. If there is a denominator of zero, try to use the Laws of Limits or factor
Strategy For Calculating Limits: 2. Graph it
Graph the function, use calculator to see if you can find value for ‘a’. If there is an error, factor the problem out or use Laws of Limits
Strategy For Calculating Limits: 3. Factor
Remove common factors by using Algebra, if you have a root, try to use the conjugate to work it out. Then plug in ‘a’ for X values
Strategy For Calculating Limits: 4. Laws of Limits
Try to apply the Laws of limits.
Removable Discontinuity
A hole in the graph of a function. Generally, if you factor out a problem or can cancel out an x value, this is where the removable continuity is
x + 5/(x-5)(x+5)
x + 5 means x = -5, removable discontinuity at -5
Limits moving through compositions
lim f(g(x)) = f lim g(x) if f is continuous
Intermediate Value Theorem
If function f is continuous on the integral from A to B inclusive, and N is any number between f(a) and f(b), assuming they aren’t equal to each other, then there exists ‘c’ in the interfal from A to B such that f(c) = N
tl;dr- if you set a section AB of f(x), there is a number ‘N’ between f(a) and f(b) that is called c
Limit of a Polynomial/Constant towards positive infinity
moves towards infinity
(lim 2x as x –> infinity) = infinity
(lim x^2 +2x+5 as x –> infinity) = infinity
Limit of a Polynomial/Constant towards negative infinity
moves towards infinity
(lim 2x as x –> -infinity) = -infinity
(lim x^2 +2x+5 as x –> -infinity) = infinity
Limit of Constant
lim 3 as x –> infinity = 3
Rational Expression (Numerator Power < Denominator Power)
limit is 0
Rational Expression (Numerator Power = Denominator Power)
Limit is ratio of leading coefficients
8x^2 +3x
Rational Expression (Numerator Power > Denominator)
Limit is infinity
