Unit 1: Limits and Continuity (copy)

Limits are the value that a function approaches as the variable within the function gets nearer to a particular value.
We don’t really care what’s happening at the point, we care about what’s happening around the point
To find the limit of a simple polynomial, plug in the number that the variable is approaching

Look on a graph to see what it approaches * If the graph approaches two different values for the same number, the limit does not exist
Estimate from a table
Algebraic Properties

Algebraic Manipulation * You can factor the numerator and denominator, then cancel any removable discontinuities * This is mostly useful if you get limits where the denominator is equal to 0
For example, (x+3)(x+2)/(x+3)(x-3) * (x+3) is able to be removed → removable discontinuity

Conditions * For all values of x in the interval that contains a, g(x) ≤ f(x) ≤ h(x) * g and h have the same limit as x approaches a
lim g(x) = L, lim h(x) = L, therefore lim f(x) = L
Trig limits as x approaches 0: * lim [sin(x)/x] = 1 * lim [(cos(x)-1)/x] = 0 * lim [sin(ax)/x] = a * lim [sin(ax)/sin(bx)] = a/b

Jump Discontinuity * Occurs when the curve “breaks” at a particular place and starts somewhere else * The limits from the left and the right will both exist, but they will not match
Essential/Infinite Discontinuity * The curve has a vertical asymptote
Removable Discontinuity * An otherwise continuous curve has a hole in it * “Removable” because one can remove the discontinuity by filling the hole
Continuity Conditions * For f(x) to be continuous when x=c: * f(c) exists * the limit as x→c exists * lim f(x) = f(c) * x→c
A function is continuous on an interval if it is continuous at every point on that interval

Removing Discontinuities

You can remove a discontinuity by redefining the function without that point in the domain
This is frequently done by factoring out a common root between the numerator and denominator

\ Limits and Asymptotes

Vertical asymptote: a line that a function cannot cross because the function is undefined there
Horizontal asymptote: the end behavior of a function * A horizontal asymptote can be crossed

\ Horizontal Asymptote Rules

If the highest power of x in a rational expression is in the numerator, then the limit as x approaches infinity is infinity: there is no horizontal asymptote
If the highest power of x is in the denominator, then the limit as x approaches infinity is zero and the horizontal asymptote is the line y=0
If the highest power is the same, then the limit is the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator

\ Intermediate Value Theorem

Guarantees that if a function f(x) is continuous on the interval [a,b] and C is any number between f(a) and f(b), ten there is at least one number in the interval [a,b] such that f(x) = C