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lim (k × f(x))
k × lim f(x)
lim (f(x) ± g(x))
lim f(x) ± lim g(x)
lim (f(x) × g(x))
lim f(x) × lim g(x)
lim f(x) / lim g(x); if lim g(x) ≠ 0
lim x→0 (sinx/x)
lim x→∞ (sinx/x)
lim x→0 (sin(1/x))
does not exist
lim x→0 ((cosx - 1)/ x)
lim n→∞ (1 + (1/n))ⁿ
lim x→a (f(x)) does not exist
If lim x→a⁻ (f(x)) ≠ lim x→a⁺ (f(x)), then.....
f is continuous at x = a
f has an infinite discontinuity at x = a
lim x→a⁻ (f(x)) or lim x→a⁺ (f(x)) is ±∞
f has a jump discontinuity at x = a
lim x→a⁻ and lim x→a⁺ (f(x)) both exist, but are not equal
f has a removable discontinuity at x = a
lim x→a (f(x)) exists, but does not equal f(a)
lim x→c (g(x)) = L
If f(x) ≤ g(x) ≤ h(x) and lim x→c (f(x)) = lim x→c (h(x))=L , then......
there exists a c in [a, b] such that f(c) = k
If f is continuous on [a, b] and if k is between f(a) and f(b), then.....
If f is continuous on the closed interval [a, b], then.....
f has a maximum and minimum value in the interval [a, b]
lim x→±∞ (f(x)) = k
The graph of f has a horizontal asymptote at y = k
lim x→a (f(x)) = ±∞
The graph of f has a vertical asymptote at x = a
lim n→∞ (ⁿ√n)
lim (f(x)/ g(x)) =
lim (f'(x)/ g'(x)), for indeterminate forms 0/0 and ∞/∞
Indeterminate forms (quotients)
0/0 and ∞/∞
Indeterminate forms (difference)
∞ − ∞
Indeterminate forms (product)
0 × ∞
Indeterminate forms (exponential)
0⁰, 1^(∞), ∞⁰
To find the limit of a 0 × ∞ indeterminate form.....
rewrite the expression as 0/0 or ∞/∞, then apply L'Hopital's Rule
To find the limit of a 0⁰, 1^(∞), or ∞⁰ indeterminate form....
use logarithms to rewrite the expression as a product 0 × ∞, then rewrite as 0/0 or ∞/∞ and apply L'Hopital's Rule
Definition of Continuity
At a point:
Definition of a Limit
A limit is a number that a function approaches as x approaches a value
Slope of tangent through (x, f(x))
Slope of tangent through (a, f(a))
II a II
a if a>= 0
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.
Occurs when common factors are cancelled in a rational expression.
A characteristic of a function in which the function has two distinct limit values as x-values approach c from the left and right
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
As limit if x approached infinity
Limit as x approaches 0 of (sinx/x)/(x/sinx)=
Average rate of change
Instantaneous rate of change
slope of tangent line!
f of x
A function f is differentiable at a if
Mins and maxes on f(x) become ________ on f'(X)
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
on a plane is located in terms of two numbers