Chapter 2: Properties of Limits - Vocabulary and Concepts

Contains terms and concepts from Calculus: Concepts and Applications by Paul A. Foerster as taught by Colin Suehring at McFarland High School

indeterminate form

0/0 or infinity/infinity

limit

the value that a function approaches as the input approaches some value

removable discontinuity

a characteristic of a function in which the function is continuous everywhere except for a hole at x=c

asymptote

a line that continually approaches a given curve but does not meet it at any finite distance

step discontinuity

if f(x) approaches different numbers from the right and from the left as x approaches c, then there is a step discontinuity at x=c

limit of a product

the product of the limit of each function

limit of a sum

the sum of the limits

limit of a quotient

quotient of limits

limit of a constant

the constant times the limit

limity of the identity function (limit of x)

limit of x as x approaches c = c

limit of a constant function

that constant

continuity at a point

function f is continuous at x=c if and only if

• f(c ) exists

• lim x—> c exists

• if f(c ) = lim x—> c

continuity on an interval

function f is continuous on an interval of x-values if and only if it is continuous at each value of x in that interval

cusp

the point on the graph at which the function is continuous but the derivative is discontinuous, an abrupt change in direction or sharp point

one-sided limit

a limit that differs depending whether it is approached from left (x-->c-) or from right (x-->+)

if lim x→a⁺ f(x) ≠ lim x→a⁻ f(x) then

lim x→a f(x) does not exist

the limit does not exist and graph forms an asymptote when

lim f(x) as x approaches c -= +/- ∞

f(x) = c is a horizontal asymptote when

lim f(x) as x approaches +/- ∞ = c

the limit does not exist when

lim f(x) as x approaches +/- ∞ = +/- ∞

f(x) = c is a vertical asymptote when

lim |f(x)| as x approaches c = ∞

the intermediate value theorem

If function f is continuous for all x in the closed interval [a,b], and y is a number between f(a) and f(b), then there is a number x=c in (a,b) for which f(c)=y