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Definition: Limit of a Function
 The function f has the limit L as x approaches the number a, written

 lim f(x) = L
 x>a
if the value of f(x) can be made as close to the number L as we please by taking x sufficiently close to (but not equal to) the number a

Righthanded limit of a function
Limit as x approaches a from the right

Lefthanded limit of a function
Limit as x approaches a from the left

Limits at infinity
 lim f(x)
 x>

a nonzero number
0
= infinity

the Intermediate value theorm
If f is a continuous function on a closed interval [a,b] and M is any number between f(a) and f(b) then there is at least one number 'c' in [a,b] such that f(c)=M

Existence of Zeros in a continuous function
If f is a continuous function on a closed interval [a,b], and if f(a) and f(b) have opposite signs, then there is at least 1 solution of the equation f(x)=0 in the interval [a,b]

Secant line
Passes through at least 2 points on a curve

Tangent Line
a straight line that "just touches" the curve at a given point



Determine the equation of the tangent line of f(x)=x^{2}2x+1 at the point (0,1)
 A. Find Derivative: f'(x)=2x2
 B. Determine slope of tan line (M) at the point (0,1): f'(0)=2
 C. determine equation of tan line at the point(0,1): yy_{1}=M(xx_{1}) M=2, x_{1}=0, y_{1}=1 y=2x+1

Average Rate of Change
 Slope of the Secant Line
 f(x+h)f(x)
 h

Instantaneous Rate of Change
 Slope of the tangent line
 lim f(x+h)f(x)
 h>0 h

Product Rule
(fg)'=f · g' + g · f'

Quotient Rule
 f(x) = g(x) · f'(x)  f(x) · g'(x)
 g(x) [g(x)]^{2}

