AP Calculus AB Unit 1 (Limits)

lim x—c f(x) only exists if

the limit from the left is equivalent from the limit of the right

Removable Discontinuity

when a point is removed from a graph. It is removable bcoz f© DNE. The graph still has a limit.

If there is another point defined, the function is removable because the limit does not equal f©

Non Removable discontinuity

the limit does not exist because of unbounded behaviour (there is a vertical asymptote)

Non-removable / Jump discontinuous

a discontinuity where the left and right limits exist but are not equal, (the function jumps, thereby creating a break in the graph) the limit DNE.

m<n

lim approaches +- infinity f(x) = 0

m>n

lim appproaches +- infinity f(x) = positive or negative infinity

m=n

limit as x approaches infinity f(x) = a/b where a and b are polynomials of the same degree

limit approaches c f(x) = ?

substitute c into f(x) to find the limit value. If the value is indeterminate then simplify the equation, then substitute c until the value is not indeterminate

pennywise function

substitute c into f(x) if the value from both functions are equal, then the limit exists and is equal to that value. If not then the limit DNE bcoz lim from left doesnt equal limit from right.

limit of any trig except sin(x)/x and (1-cosx)/x

Oscillating behaviour.

squeeze theorem

if the two limits other than f(x) equal each other, than f(x) also equals that, always USE LIMIT when solving.

IVT explanation

therefore by the IVT theorem, there exists at least one “c” in (a,b) such that f© = …

continuous function

lim is continuous if

1. f of c is defined

2. limit as x approaches c of f(x) exists

3. limit as x approaches c of f(x) = f of c

f and g are continuous at x =c, which means the following are also continuous

1. kf (scalar multiple)

2. fg

3. f/g (g©) cant equal 0

4. f +- g

if g is continous at x =c , and f is continuous at g©, then

f of g (x) = f(g(x)) is continuous at x =c

limit from the left or right when approaching infinity

do not say unbounded, say approaches infinity, plug in a number from the left of the x approx. and determine whether negative or positive, do the same with right, this will tell you neg o pos.

limit overall approaching infinities

unbounded behaviour, DNE

trig function that equal smth in limits

sin(x)/x = 1

1-cos(x) / x = 0

graphs to remember

squareroot of (a²-x²). (semicircle, with a as radius

1/x² both ends pointing upward, with va at 0

absol.x / x = one end going one way, another end going the other, different y values.

log - curve (graph crosses at x=1) towards right side

x2 = parobla

rational = bottom curve going left, top curve going right

expo. growth, one curve going right,

expo decay one curve going leftt.

odd root graph = one curve shooting from origin to up

if lim approaching x axis, what to focus on

focus on the highest degree and coefficient, if the degree in num is greater than denom, substitute the infinity that is being approached and choose pos or neg infinity.

if one forgets the horizontal asymptote rules

divide everything by x to the highest degree in the denominator, then evaluate the limit as x approaches infinity.

infinity +- other number

still infinity.

lim x —c (f(x) + g(x)) = infinity

same for one sided limits and if it is negative infinity instead of infinity

if infinity and another number multiply,

equals infinity if number is positive. if negative, equals negative infinity.

lim, x—c ( f(x) times g(x) )

same for one sided limits and if it is negative infinity instead of infinity

if another number is divided by infinity

equals 0 because infinity is so big it makes the other number approach very close digits to zero.

lim x —c ( g(x) / f(x) ) = 0

same for one sided limits and if it is negative infinity instead of infinity

any number over zero

if number positive, equals infinity

if negative equals negative infinity.

and, if lim x —- +- infinity, c/x^n = 0

lim x —- +- infinity, c/x^n = 0

if f(x) gets very close to L for x>m and M is positive, limit as x approaches infinity, f(x) = L.

if f(x) gets very close to L

lim x —- +- infinity, c/x^n = 0

if f(x) gets very close to L for x<m and M is negative, limit as x approaches negative infinity, f(x) = L.