AP Calculus AB Unit 1: Limits and Continuity

Horizontal Asymptote

The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either limx→∞ f(x) = b or limx→-∞ f(x) = b.

sin0

0

sin(π/6)

1/2

sin(π/4)

1/√2 = √2/2

sin(π/3)

√3/2

sin (π/2)

1

sinπ

0

sin(3π/2)

-1

sin2π

0

cos0

1

cos(π/6)

√3/2

cos(π/4)

1/√2 = √2/2

cos(π/2)

0

cosπ

-1

cos(3π/2)

0

cos2π

1

tan0

0

tan(π/6)

1/√3 = √3/3

tan(π/4)

1

tan(π/3)

√3

tan(π/2)

undefined

tanθ (in terms of sine and/or cosine)

sinθ/cosθ

cotθ (in terms of sine and/or cosine)

1/tanθ

cscθ (in terms of sine and/or cosine)

1/sinθ

secθ (in terms of sine and/or cosine)

1/cosθ

An odd function is...

... symmetric with respect to the origin, like y=x^3, y=sinx, or y=tanx. f(-x) = -f(x)

graph of y = sinx

graph of y = tanx

graph of y = cosx

graph of y = secx

graph of y = 1/x

graph of y =√x

graph of y = √1-x^2

graph of y = |x|

graph of y = lnx

graph of y = e^x

An even function is...

...symmetric with respect to the y-axis (like y = x^2, y = cosx, or y = |x|

ƒ(-x) = -ƒ(x)

A horizontal asymptote occurs at y = b when...

... lim ƒ (x) = b

x →±∞

A vertical asymptote occurs at x = a when...

... lim ƒ ( x ) = ±∞

x→a⁺⁻

indicated by substitution yielding ƒ(a) = nonzero/0

ƒ(x) is continuous at x = c when...

1. ƒ(c) exists

2. lim ƒ(x) exists; and

x→c

3.lim ƒ(x) = f(c)

x→c

Intermediate Value Theorem: If f is continuous on [a,b], then...

... there exists some c in [a,b] such that f(c) = k for any y-value k between f(a) and f(b)

Average rate of change is...

∆y/∆x = (ƒ(b) - f(a)) / (b-a)

A secant line is...

..the line connecting two points on a curve.

Instantaneous rate of change is...

lim (ƒ(a+h) - ƒ(a)) / h

h→0

A tangent line is...

...the line through a point on a curve with slope equal to the slope of the curve at that point.

A normal line is...

... the line perpendicular to the tangent line at the point of tangency.

Point-slope form of a linear equation

y-y₁ = m(x-x₁)

Verbal Definition of a Limit

The y-value that a function approaches as the x-value gets closer and closer to a point c from left and right.

How can a limit exist at a point where a function value does not exist?

Where a hiccup function exists, where there is a hole at c, but the graph approaches the same y-value as it approaches c from the left and right.

How can a function value exist at a point where a limit does not exist?

If c does not approach the same y-value from both side, then the limit does not exist, but there is a function value when approaching c from either left or right.

Three cases in which a limit DNE (does not exist):

1. differing behavior from left and right

2. unbounded behavior (limit approaching infinity)

3. oscillating behavior

What is an indeterminate form and what does it indicate?

0/0 is called an indeterminate form that is yielded when using direct substitution to find a limit value and indicates that there is a hole in the graph. Therefore, another method must be used to find the limit.

The Sandwich Theorem

If f(x) ≤ g(x) ≤ h(x) for all values where x ≠ c in some interval about c, and limx→c f(x) = limx→c h(x) = L, then limx→c g(x) = L.

- therefore all limits have the same value at y values at c

If direct substitution yields the indeterminate form 0/0, try:

1. simplifying complex fractions

2. factor (trnomial, GCF, difference of two squares)

3. expanding a binomial power

Pascal's Triangle

Why do we only care about certain terms in the numerator and denominator?

b/c the terms with the highest degrees in the denominator and numerator increase the fastest

End behavior and asymptotes when the larger degree is in the denominator?

for rational functions f(x) = ax^m+... and g(x) = bx^n +...

End Behavior: limx→±∞ f(x)/g(x) = 0

Horizontal Asymptote: y = 0

End behavior and asymptotes when the largest degrees in the denominator and numerator are equal?

for rational functions f(x) = ax^m+... and g(x) = bx^n +...

End Behavior: limx→±∞ f(x)/g(x) = a/b

Horizontal Asymptote: y = a/b [the ratio of leading coefficients]

End behavior and asymptotes when the largest degree is in the numerator?

for rational functions f(x) = ax^m+... and g(x) = bx^n +...

End Behavior: limx→±∞ f(x)/g(x) = DNE

Horizontal Asymptote: no horizontal asymptotes

What must be true about a function if it has a horizontal asymptote?

The limit as x approaches ±∞ exists.

If a function has a horizontal asymptote y = k, what must be true about this function?

The limit as x approaches ±∞ of f(x) exists.

List these functions from the slowest relative growth rate to the fastest relative growth rate based on f(100) [explain why]:

cosx

a^x , where a > 1

x^n, where n > 0

logx

1. cosx [b/c the range of cosx always stays between -1 and 1

2. logx [b/c this logarithms always produces a gradual concave down slope]

3. x^n [b/c a polynomial is only ever increasing based on a constrained exponent, so x is the variable]

4. a^x [b/c an exponential is continuously compounded based on a variable exponent x, so as the exponential approaches infinity, the exponential function will always be greater than the polynomial functions; however it is important to note that an exponential function, at some point, may be less than a polynomial]

How can a function have more than one horizontal asymptote?

If the value of the limit as x approaches -∞ of f(x) is different from the value of the limit as x approaches +∞ of f(x).

Why is lim x→-∞ e^x equivalent to lim x→∞ 1/e^x?

b/c e^-∞ = 1/e^∞

Vertical Asymptote

line x = a is a vertical asymptote of the graph of a function y = f(x) if either limx→a⁺ f(x) = ±∞ or limx→a⁻ f(x) = ±∞

-occurs when a value of x makes the denominator 0

-0/0 means there's a hole in the graph

-nonzero/0 means a vertical asymptote (limit DNE)

How can we tell whether the function approaches +∞ or −∞ at a vertical asymptote?

1. find which value(s) of x makes the denominator zero and the value(s) of x that makes the numerator zero

2. put the zeroes on a line and test the direction of the graph on the left and right of the x value that makes the denominator zero

3. the sign (positive or negative) of evaluated number on the graph towards the left and right of the x value making the denominator zero tells you whether the function if approaching ±∞

- if the graph behavior towards the left/right if negative, then the graph is approaching -∞ and vice-versa