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Level 602

Limits & Continuity AB & Bc II


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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
Another name for a linear approximation to the curve is a
limit
The unique value that a function approaches as x-values of the function approach c from the left and right sides
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
f(x)=1/x²
Example: Infinite Limit v Limits at Infinity
Asymptote
An imaginary line on a graph that acts as a boundary line.
vertical asymptote
comes from a factor in the denominator that doesnt cancel out
horizontal asymptote
a line that the curve approaches as x goes to infinity but never reaches
Infinite Limits at Infinity
If f(x) becomes arbitrarily large as x becomes arbitrarily large, then lim x→∞ f(x)∞
End Behavior
Look at HA
End Behavior of Powers and Polynomials
Find the term with x with the highest power
End Behavior of Rational Functions
Divide numerator and denominator by x^n where n is the largest power applied to x in the denominator
Values oscillate between ∓1 as x→∓∞
End Behavior of sin & cos x
Continuity
Refers to whether a function contains holes, jumps, or breaks
Continuity at a point
A function f is continuous at a if lim x→a f(x) = f(a)
Continuity Checklist
In order for f to be continuous at a
Continuity & Composite Function Rules
If functions f and g are continuous at a, then the following are also continuous at a, assuming that c is a constant and n > 0 is an integer
Continuity on an Interval
A function is continuous on an interval if it is continuous at every point in that interval.
[f(x)]^n/m
Continuity of Functions with Roots
Intermediate Value Theorem
If f(a) > 0 and f(b) < 0 then there is a real zero between a and b
Removable
Discontinuity Classes
ͼ
Epsilon
σ
Lambda
Limit Proximity
lim x→a f(x)=L means that f(x) is near to L for all x near to a
f(−x) = f(x)
f(x) is an even function if
f(−x) = −f(x)
f(x) is an odd function if
Limit Existence Theorem: [limit as x→c of f(x)] = L if and only if
[limit as x→c⁺ of f(x)] = L and [limit as x→c⁻ of f(x)] = L
b
limit as x→c of b =
c*[limit as x→c of f(x)]
limit as x→c of c*f(x) =
limit as x→c of [f(x)*g(x)] =
[limit as x→c of f(x)] * [limit as x→c of g(x)]
limit as x→c of [f(x)/g(x)] =
[limit as x→c of f(x)] / [limit as x→c of g(x)]
[limit as x→c of f(x)]ⁿ
limit as x→c of [f(x)]ⁿ =
0
[limit as θ→0 of (sin θ)/θ] =
1
[limit as θ→0 of (1 − cos θ)/θ] =
[limit as x→c of f(x)] = f(c)
f(x) is continuous at x = c if