Feb 4 - Feb 8 Flashcards
Continunity 2.5 Limits at Infinity 2.6 Derivatives and Rates of Change 2.7
Intuitive Definition of a Limit at Infinity
Let f be a function defined on some interval (a, infinity) then
(as x approaches infinity)
lim f(x) = L or f(x) –> L
means that the values f(x) can be made arbitrarily close to L by requiring x to be sufficiently large positive, and this is the opposite for x approaching negative infinity
Horizontal Asymptote
The Line of y = L is horizontal asymptote of the curve y = f(x) if either
(x approaching infinity)
lim f(x) = L
(x approaching negative infinity)
lim f(x) = L
If r > o is a rational number then
(as x approaches infinity)
lim 1/x^ r = 0
If r > 0 is a rational number such that x^ r is defined for all x then
(as x approaches negative infinity)
lim 1/x^ r = 0
Infinite Limits at Infinity
The notation
(x approaches infinity)
lim f(x) = infinity
is used to indicate that the values of f(x) become large as x becomes large. Similar meanings are attached to the following symbols:
(x approaches negative infinity)
lim f(x) = infinity
lim f(x) = - infinity
(x approaches infinity)
lim f(x) = - infinity
Precise Definition of a Limit at Infinity
Let f be a function defined in some interval (a, infinity) then
(x approaches infinity) lim f(x) = L
means that for every E > 0 there is a corresponding number N such that
if x > N then |f(x) - L| < E
Definition of an Infinite Limit at Infinity
LEt f be a function defined on some interval (a, infinity) then
(x approaches infinity)
lim f(x) = infinity
means that for every positive number M there is a corresponding positive number N such that
if x > N then f(x) > M
Tangents
If a curve C has an equation y = f(x) to compute the slope of the secant line PQ
mPQ = f(x) - f(a) / x -a
Tangent Line Slope
m = lim f(x) - f(a) / x - a
as x approaches a
or
m = lim f(a + h) - f(a) / h
as h approaches 0
The Average Velocity Over This Time Interval is
avg velocity = displacement / time = f(a + h) - f(a) / h
Equation for Instantaneous Velocity for limits
v(a) = lim f(a + h) - f(a) / h
as h approaches 0
Derivative of a function f at a number a, denoted by f’(a) is
f’ (a) = lim f(a + h) - f(a) / h
as h approaches 0
