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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