
Deffiniation of a Limit
Let f be a function defined on an open interval containing c (exept possibly at c) and let L be a real number
 means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, it can be stated that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.

commontypes of behaviors associated with the nonexitence of a limit
 1. f(x) approaches a different number from the right side of c than it approaches from the left side.
 2. f(x) increases or decreases without bound as x approaches c
 3. f(x) oscillates between two fixed values as x approaches c

What are three basic limits

Properties of limits: Sum or difference

Properties of limits: Product

Properties of limits: Quotient
and

Properties of limits: Power
and

Limit of a composite Function
If f is continuous at b and lim _{x→a} g(x) = b
then lim x→a f(g(x)) = f (lim x→a g(x)) = f (b).

Limit of Trigonometric Functions:
find lim_{x→c} for sin(x) ; cos (x) ; sin(x)/x ; (1cos(x))/(x)

Example : Evaluate
 Substituting 0 for x yields 5/0, which is meaningless; hence, = DNE. (Remember, infinity is not a real number.)
 THIS IS AN ASYMPTOTE, NONREMOVABLE DISCONTINUITY (numerator ≠ 0 , denominator = 0)

A stradegy For finding limits
 1. Learn to recognize which limits can be evaluated by direct substituation
 2. If the limit of f(x) as x approaches c cannot be evaluated by direct substituaiton, try to find a afunction g that agrees with f for all of x other than x=c
 3. apply the theorem for removable discontinuities such that; lim_{ x→c} f(x) = lim _{x→c} g(x) = g(c)
 4. use a graph or table to reinforce conclusion

Functions that agree at all but one point
let c be real and let f(x) = g(x) for all x ≠ c in an open interval containing c
If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists
lim _{x→}_{c} f(x) = lim _{x→}_{c }g(x)


The existence of a limit
let f be a function and let c and L be real numbers

Definition of continuity on a closed interval
A function is said to be continuous on [a,b] if and only if
 1. it is continuous on (a,b),
 2. it is continuous from the right at a and
 3. it is continuous from the left at b.

properties of continuity
1. scalar multiple:
2. sum or difference:
3. Product:
4. quotient:
 1. scalar multiple: bf
 2. sum or difference: f (+  ) g
 3. Product: fg
 4. quotient: f/g, if g(c) ≠ 0

Continuity of a compsoite function
If the function g is continuous at a number a, and f is continuous at g(a), the composite function _{} is also continuous at a.

intermediate value theorem
If a function f is continuous on a closed interval [a,b], then for every value k between f(a) and f(b) there is a value c between a and b such that f(c) = k.

definition of infinite limits
We say if we can make f(x) arbitrarily large for all x sufficiently close to x=a, from both sides, without actually letting .
We say if we can make f(x) arbitrarily large and negative for all x sufficiently close to x=a, from both sides, without actually letting .

Definition of a vertical asymptote
if f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f

finding vertical asymptote
 let h and g be continuous on an open interval containing c. If g(c) ≠ 0, h(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by:

 has a vertical asymptote at x = c

properties of infinite limits: sum or difference
Given the functions and suppose we have,
for some real numbers c and L.

Properties of Infinite Limits: Product
Given the functions and suppose we have,
for some real numbers c and L. Then,

Properties of infinite limits: Quotient
Given the functions and suppose we have,
for some real numbers c and L. Then,

Definition of a tangent line with slope m
also known as the slope of the graph of f at x = a
if f is defined on an open interval containing a, and if the limit: exists, then the lines passing through (a, f(a)) with slpoe m is the tangent line to the graph of f at the point (a, f(a))

Definition of the derivative of a function
 the derivative of f at x

differentialbility implies continuity
if f is differentiable at x = c, then f is continuous at f = c

The constant rule of the derivative
the derivative of a constant function is 0. That is, if c is a real number, then

the power rule of the derivative
if n is a rational number, then the function f(x) = x ^{n} is differentialble. For f to be differentiable at x=0 then n must be a number such that x ^{n1} is defined on the interval containing 0 _{
}

The sum and differnece rules of the derivative
 The derivative of the sum of two functions is the sum of the derivatives of the two functions:
 .
 Likewise, the derivative of the difference of two functions is the difference of the derivatives of the two functions.

the derivative of the sin and cos function

Alternate limit form of the derivative

How to find the equation of the tangent line through a point (x_{1},y_{1})
find the derivative f'(c) = m = slope
plug m into the point slope formula yy_{1}=m(xx_{1})

