Limits and Continuity

under what conditions does a finite limit exist?

only if the left and right hand limits both approach the same value

3 step definition for continuity at a point (a function f(x) is continuous at a point x=c if...)

1. f(c) exists

2. lim x->c f(x) exists

3. lim x->c f(x) = f(c)

criteria for removable point discontinuity

2 is satisfied (lim x->c f(x) exists)

either 1 or 3 are not satisfied or both are not satisfied

criteria for nonremovable jump discontinuity

the one sided limits are not equal at x=c, regardless of the value of f(c)

if the function is continuous at that POINT, how do you evaluate the limit?

just plug it in

definition of nonremovable infinite discontinuity (VA) - "essential"

if at x=c it approaches either positive or negative infinity

oscillating discontinuity

3 cases where a limit DNE

1. jump discontinuity

2. unbounded behavior

3. oscillating behavior

if you try to evaluate and you get 0/0 what should you do?

factor to remove the discontinuity, or use conjugate, or use trig identities, then evaluate for the limit

if the limit approaches positive or negative infinity, does it technically exist?

no, limit DNE even though it can be described (because a limit only "exists" if it converges to a finite, real number)

for data in a table, do not...

assume anything other than you are given

under what condition do the 6 properties of limits apply?

if lim x->c f(x) exists, if lim x->c g(x) exists

for limits of composite functions, what must I include? (or else will be penalized!!)

when finding L, must include direction (either + or -)

squeeze theorem conditions

add: the inequality is true for all x, except possibly at x=a (because whether or not g(x) is defined at x=a does not affect that the limit is true)

list functions by how fast they grow for large x in increasing order (explain for k and x!)

k -> log x -> xⁿ -> bˣ -> x! -> xˣ

note 1: k is a constant

note 2: for x!, x is a whole number

lim x->0 (sin x)/x

lim x->0 (1-cos x)/x

are the graphs of |x|/x and x/|x| the same?

yes