AP Calculus Unit One

Average velocity 

Slope of secant line

Instantaneous velocity

Slope of tangent

lim f(x) = L

x→a

The limit of f(x), as x approaches a, equals L

lim f(x) = L

x→a⁻

The limit of f(x), as x approaches a from the left, equals L

lim f(x) = L if and only 

x→a

lim f(x) = L and lim f(x) = L

x→a⁻                x→a⁺

lim [f(x) + g(x)]

x→a

lim f(x) + lim g(x) 

x→a        x→a

lim [f(x) - g(x)] = 

x→a 

lim f(x) - lim g(x) 

x→a      x→a

lim [cf(x)] = 

c lim f(x)

   x→a

lim [f(x)g(x)]

x→a

lim f(x) * lim g(x) 

x→a      x→a

lim [f(x) / g(x)] =

x→a

lim f(x) / lim g(x) if lim g(x) ≠ 0 

x→a      x→a        x→a

lim [f (x)]ⁿ = 

x→a

[lim f (x)]ⁿ 

x→a

lim c = 

x→a

c

lim x

x→a

a

lim xⁿ =

x→a

aⁿ where n is a positive interger

lim ⁿ√x =

x→a

ⁿ√a

lim ⁿ√f(x) =

x→a

ⁿ√lim f(x) 

x→a

If f is a polynomial or a rational function and a is in the domain of f, then 

lim f(x) = f(a)

x→a

If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist as a x approaches a, then

lim f(x) ≤ lim g(x) 

x→a       x→a

The squeeze theorem

If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and

lim f(x) = lim h(x) = L   

x→a       x→a            then

lim g(x) = L 

x→a

A function f is continuous at a number a if

lim f(x) = f(a)

x→a

A function f is continuous from the right at a if 

lim f(x) = f(a) and f is continuous from the left a if lim f(x) = f(a)  

x→a⁺                                                                     x→a⁻

A function f is continuous on an interval if

it is continuous at every number in the interval.

If f and g are continuous at and c is a constant, then the following functions are also continuous at a:

1. f + g

2. f - g

3. cf

4. fg

5. f/g if g(a) ≠ 0

Any polynomial is continuous everywhere; that is, it is continuous on

Domain (-∞, ∞)

Any rational function is continuous wherever it is defined

that is, it is continuous on its domain.

The following types of functions are continuous at every number in their domains

polynomials, rational function, root functions, trigonometric functions, inverse trigonometric functions, exponential functions and logarithmic functions.  

If f is continuous at b and lim g(x) = b, then lim f(g(x)) = f(b). 

                                         x→a                     x→a

In other words, lim f(g(x)) = f (lim g(x))

                         x→a               x→a

If g is continuous at a and f is continuous at g(a), then

The composite function f of g given by (f of g)(x) = f(g(x)) is continuous at a.

The intermediate value theorem

Suppose that f is continuous on the closed interval [a, b] and let N any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a,b) such that f(c) = N.

lim tan^-1 x =

x→ - ∞

-π/2 

lim tan^-1 x =

x→∞

π/2 

lim e^x =

x→ -∞

0