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Level 516

Limits & Continuity


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Continuous Function
A function that can be graphed without breaks, holes, or gaps
Discontinuous Function
A function that is not continuous; 3 types
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
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
Removable discontinuity
Occurs when common factors are cancelled in a rational expression.
Nonremovable Discontinuity
Describes infinite and jump discontinuities because they cannot be eliminated by redefining the function at that point
End Behavior
Look at HA
limit
The unique value that a function approaches as x-values of the function approach c from the left and right sides
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
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.
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
Instantaneous Rate
The rate of y to x for a given point along the x axis
Secant Line
a line that intersects a circle at two points on the same plane
Tangent Line
A line that only touches one point on a curve
Limit Notation
(the arrow notations is a subscript under lim that I can't do here)
Two-sided limit
One in which the limit is the same whether it is approached from above or below
One-sided limit
a limit that differs depending whether it is approached from above or below
Limit Laws: Sum
Assume lim f(x) and lim g(x) exist for x→a
Limit Laws: Difference
Assume lim f(x) and lim g(x) exist for x→a
Limit Laws: Constant Multiple
Assume lim f(x) and lim g(x) exist for x→a
Limit Laws: Product
Assume lim f(x) and lim g(x) exist for x→a
Limit Laws: Quotient
Assume lim f(x) and lim g(x) exist for x→a
Limit Laws: Power
Assume lim f(x) and lim g(x) exist for x→a
Limit Laws: Fractional Power (or, Root)
Assume lim f(x) and lim g(x) exist for x→a
Squeeze Law
If f(x)≤g(x)≤h(x) for values of x near a, except possibly at a, then
Infinite Limit
If f is defined for all x near a, and f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a, then
Limits at infinity
The dependent variable approaches a finite number as the independent variable becomes arbitrarily large