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Level 517

Limits & Continuity II


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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
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)/g(x)
lim f(x) / lim g(x); if lim g(x) ≠ 0
1
lim x→0 (sinx/x)
0
lim x→∞ (sinx/x)
lim x→0 (sin(1/x))
does not exist
0
lim x→0 ((cosx - 1)/ x)
e
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(a) exists
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
1
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
differential calculus
calculus arising from the tangent line problem
integral calculus
calculus arising from the area problem
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
the relationship between one-sided and two-sided limits
if the one-sided limits from the left and right sides of a are equal then L is the same for the two-sided limit
infinite limits
when x approaches a, f(x) increases or decreases without bound
Vertical Asymptotes
zeroes of denominator
Limit of Polynomials Theorem
For p(x) = c0 + c1x^1 + ... + cnx^n,
indeterminate form of type 0/0
a quotient f(x)/g(x) in which the numerator and denominator both have a limit of zero as x -> a
Limits of Rational Functions Theorem
Let f(x) = p(x)/q(x) be a rational function, and let a be any real number. If q(a) =/= 0, then lim f(x) = f(a).
limits of piecewise functions
the two-sided limits of these functions are found by first finding the one-sided limits
Limit Laws for Limits at Infinity
these laws are the same as the basic limit properties
Limits of x^n as x -> +/- infinity
the limit of x^n as x -> +infinity = +infinity for all n
Limits of Polynomials as x -> +/- infinity
The end behavior of a polynomial matches the end behavior of its highest degree term
Finding Limits of Rational Functions as x -> +/- infinity
The end behavior of a rational function matches the end behavior of the quotient of the highest degree term in the numerator divided by the highest degree term in the denominator
Reference Limits
lim ln x = +infinity, lim e^x = +infinity
Simple Limit Definition
Let f(x) be defined for all x in some open interval containing the number a, with the possible exception that f(x) need not be defined at a. We will write
-f(c) is defined
Rules of of being continuous
Rules of being continuous on an interval
- f is continuous on (a, b)
Properties of continuous functions
- [f + g] is continuous at c
Continuity of Polynomials and Rational Functions
- a polynomial is continuous everywhere
Continuity of Compositions
lim f( g(x) ) = f( lim g(x) )
Intermediate Value Theorem - Approximating Roots
If f is continuous on [a, b], and if f(a) and f(b) are nonzero and have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval (a, b)
Continuity of Trig Functions
lim sin x = sin c
Continuity of Inverse Functions
If f is a one-to-one continuous function defined on an interval, then its inverse function f^-1(x) is also continuous
Exponential Function Continuity Theorem
-The function b^x is continuous on (-inf., +inf.)
The Squeezing Theorem
g(x) <_f(x) <_ h(x) for all x in some open interval containing the number c, with the possible exception that the inequalities need not hold at c. If g and h have the same limit as x -> c,
lim (sin x)/x = 1
Special Limits (illustrate the Squeeze Theorem)
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
Lines, Polynomials, sine, cosine
Name some functions that are continuous everywhere
Rationals, Radicals, tangent, cotangent, secant, cosecant
Name some functions that are continuous on their domain
f(g(x)) is continuous at x = c if
f(x) is continuous at g(c), and g(x) is continuous at c.
limit as x→c of f(x) = ±∞
f(x) has a vertical asymptote at x = c if
y = L is a horizontal asymptote of f(x) if
limit as x→+∞ of f(x) = L OR limit as x→−∞ of f(x) = L