AP Calc AB unit 1

One sided limits…

Evaluate what the function is approaching from their respective direction

Two sided limits can only exist if

The one sided limits are equal

limits fail to exist at what discontinuities?

Vertical asymptotes (essential discontinuity)

jump discontinuities

Removable point discontinuity

f(x) fails to exist at a certain point— hole in graph can remove by factoring and crossing out denominator

Besides one-sided limits and certain discontinuties, when would a limit fail to exist?

if the value of the function will keep oscillating to infinity

Example: sin x will oscillate between 1 and -1 and never exceed those values

constant multiple property of limits

lim (b ∗ f(x)) = b (lim f(x))

x→c x→c

sum property of limits

lim (f(x) + g(x)) = lim f(x) + lim g(x)

x→c x→c x→c

difference property of limits

lim (f(x) − g(x)) = lim f(x) − lim g(x)

x→c x→c x→c

product property of limits

lim (f(x) ∗ g(x)) = lim f(x) ∗ lim g(x)

x→c x→c x→c

quotient property of limits

lim (g(x)/f(x)) = lim g(x) / lim f(x)

x→c x→c x→c

power property of limits

lim (f(x))^n = (lim f(x))^n

x→c x→c

root property of limits

lim n^√f(x) = √lim f(x)

x→c x→c

strategy for solving limit of polynomial

first: plug in c value

— If that gives you =0, check by factoring, cancelling out, and then plugging in again.

— If that doesn’t work, plug in numbers more or less than the c value.

— Can also try L’hopital’s rule if plugging in results in an indeterminant (0/0, plus/minus infinity / plus/minus infinity)

strategy for solving limit of rational function

factor numerator and denominator using polynomial or synthetic division, if possible. Then, find the limit by plugging in c for all values of x.

how to find limit of radical function (function w square root)

multiply by the conjugate of the function to rationalize denominator. Plug in c for x unless the nth root is positive and the radicand will be negative (this would result in a complex answer).

general rule for finding limits

same as polynomial- plug in c → factor and solve → plug in values more or less than c → can also use l’hopital’s rule

how to find limit of composite functions

(one function subbed into another)

find the limit of the inner function and plug in this R value for x in the outer function.

what is the squeeze theorem used for

evaluate limit of function bounded by two other functions

squeeze theorem

It requires three functions: f(x), g(x), and h(x).

The following must be true: f(x) ≤ g(x) ≤ h(x) over the interval surrounding but not necessarily including the point x = c.

Under these conditions, if lim f(x) = lim h(x) = L then lim g(x) = L.

x→c

discontinuity occurs at…

occurs at vertical asymptotes, jump discontinuities, oscillating behavior, and removable discontinuities.

3- part def of continuity

for an interval to be continuous…

definition of continuity fulfilled on whole interval

To be continuous on [a,b], a function: must be continuous; must have the left one sided limit equal to f(a); must have the right one-sided limit equal to f(b).

• Polynomial, power, exponential, absolute value, rational, logarithmic, and trigonometric functions are continuous on all points over their respective domains.

• If the function g is continuous over an interval and the function f is continuous over the same interval, then the composition f ∘ g is continuous over that interval.

how to remove a discontinuity

factor and cross out

infinity limits are often associated with —

vertical asymptotes

how do you approximate a limit at infinity?

plug infinity in

limit of a rational function at infinity depends on the degree of the — and —

numerator, denominator

denom higher degree: right side limit = 0

equal: ratio of leading coefficients

numerator higher degree: infinity

How do AP test graders want you to show your work on infinity limits?

divide everything by the highest degree, simplify, plug in infinity

numbers with infinity in the denominator will approach 0. follow the numerator denominator degree rules to find the answer.

lim (sin x / x) = …

x → 0

1

lim (cos x - 1 / x) = …

x → 0

0

lim (sin ax / x) = …

x → 0

a

lim (sin ax / sin bx) = …

x → 0

a / b