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lim (k × f(x))

k × lim f(x)

lim (f(x) ± g(x))

lim f(x) ± lim g(x)

lim (f(x) × g(x))

lim f(x) × lim g(x)

lim f(x)/g(x)

lim f(x) / lim g(x); if lim g(x) ≠ 0

1

lim x→0 (sinx/x)

0

lim x→∞ (sinx/x)

lim x→0 (sin(1/x))

does not exist

0

lim x→0 ((cosx - 1)/ x)

e

lim n→∞ (1 + (1/n))ⁿ

lim x→a (f(x)) does not exist

If lim x→a⁻ (f(x)) ≠ lim x→a⁺ (f(x)), then.....

*f(a) exists

f is continuous at x = a

f has an infinite discontinuity at x = a

lim x→a⁻ (f(x)) or lim x→a⁺ (f(x)) is ±∞

f has a jump discontinuity at x = a

lim x→a⁻ and lim x→a⁺ (f(x)) both exist, but are not equal

f has a removable discontinuity at x = a

lim x→a (f(x)) exists, but does not equal f(a)

lim x→c (g(x)) = L

If f(x) ≤ g(x) ≤ h(x) and lim x→c (f(x)) = lim x→c (h(x))=L , then......

there exists a c in [a, b] such that f(c) = k

If f is continuous on [a, b] and if k is between f(a) and f(b), then.....

If f is continuous on the closed interval [a, b], then.....

f has a maximum and minimum value in the interval [a, b]

lim x→±∞ (f(x)) = k

The graph of f has a horizontal asymptote at y = k

lim x→a (f(x)) = ±∞

The graph of f has a vertical asymptote at x = a

1

lim n→∞ (ⁿ√n)

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

lim (f'(x)/ g'(x)), for indeterminate forms 0/0 and ∞/∞

Indeterminate forms (quotients)

0/0 and ∞/∞

Indeterminate forms (difference)

∞ − ∞

Indeterminate forms (product)

0 × ∞

Indeterminate forms (exponential)

0⁰, 1^(∞), ∞⁰

To find the limit of a 0 × ∞ indeterminate form.....

rewrite the expression as 0/0 or ∞/∞, then apply L'Hopital's Rule

To find the limit of a 0⁰, 1^(∞), or ∞⁰ indeterminate form....

use logarithms to rewrite the expression as a product 0 × ∞, then rewrite as 0/0 or ∞/∞ and apply L'Hopital's Rule

Definition of Continuity

At a point:

Definition of a Limit

A limit is a number that a function approaches as x approaches a value

lim(h→0)=[f(x+h)-f(x)]/h

Slope of tangent through (x, f(x))

lim(x→a)=[f(x)-f(a)]/(x-a)

Slope of tangent through (a, f(a))

y-y1=m(x-x1)

point-slope form

II a II

a if a>= 0

Existence Theorem

tells you if something exists, but not how to find it

intermediate value theorem (IVT)

suppose f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b). Then there exists a number "e" such that f(e)=N.

Removable discontinuity

Occurs when common factors are cancelled in a rational expression.

Jump Discontinuity

A characteristic of a function in which the function has two distinct limit values as x-values approach c from the left and right

Infinite Discontinuity

A characteristic of a function in which the absolute value of the function increases or decreases indefinitely as x-values approach c from the left and right

Find HA

As limit if x approached infinity

1

Limit as x approaches 0 of (sinx/x)/(x/sinx)=

msec

Average rate of change

mtan

Instantaneous rate of change

Derivative=

slope of tangent line!

f(x)

f of x

f'(x)

velocity

f''(x)

acceleration

f''"(x)

jerk

f'(a) exists

A function f is differentiable at a if

zeroes

Mins and maxes on f(x) become ________ on f'(X)

concavity

a graph is concave up is the tangent lines are below the graph; a graph is concave down if the tangent lines are above the graph

when f changes concavity,

f' has a min or a max

When a graph is decreasing,

the slopes of the tangent lines are negative

a point

on a plane is located in terms of two numbers