Any E>0, there exists a d>0

0<|x-a|<d => |f(x)-L|<E

a exists in Real, I is an OPEN interval which contains a

L = lim f(x)

^x->a

exists iff f(x_n) -> L as n -> infinity

for every sequence x_n exists I\{a} which converges to a as n -> infinity

f(x)&g(x) converge as x -> a

lim(f+g)(x)

x -> a

lim f(x) + lim g(x)

x -> a

f(x)&g(x) converge as x -> a

lim (cf)(x)

x -> a

c lim f(x)

x -> a

f(x)&g(x) converge as x -> a

lim (fg)(x)

x -> a

lim f(x) lim g(x)

x -> a

f(x)&g(x) converge as x -> a

lim g(x) is non-zero

lim (f/g)(x)

x -> a

lim f(x)

lim g(x)

x -> a

a exists in R, I is an open interval containing a, and f,g,h are real functions defined everwhere on I except possibly at a

i) if g(x)<h(x)<f(x) for all x exists I\{a} and

lim g(x) = lim f(x) = L

x -> a

lim h(x) = L

ii) if |g(x)| < M for all x exists I\{a} and f(x) -> 0

as x -> a, then

lim g(x)f(x) = 0

x -> a

a exsists R, I is an open interval which contains a, & f,g are real functions defined everwhere on I except possible at a.  If f and g have a limit as x -> a and f(x)<g(x) for all x exists I\{a}

lim f(x) < lim g(x)

x -> a

Right-Handed Limits

lim f(x)

x -> a⁺

f is defined on some open interval I with left endpoint a and for every E>0 there is a d>0 such that

a+d exsists in I and

a < x < a+d => |f(x) - L|< E

Left-Handed Limits

lim f(x)

x -> a^-

lim f(x) 

x -> a

exists iff

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

x -> a⁺      x->a^-

f(x) -> L as x -> Infinity iff there exists c>0 such that

(c,Infinity) subset of Dom(f(x)), E>0 there is an M exists in R such that

x > M => |f(x) - L|

I contains a such that I\{a} subset Dom(f(x)) and given 

M exists R there is a d > 0 such that

0<|x - a|<d => f(x) > M

a exists I iff given E > 0 there is d > 0

|x - a| < d, x exists I => |f(x) - f(a)| 

given f,g continuous

f + g, fg, and cf are continuous

if g(x) is non-zero f/g is continuous

A and B are subsets of R, that f : A -> R and g : B -> R

if f(A) is a subset of B for every x exists A, then

g(f(x)) : A -> R

i) L = lim f(x)

x -> a

exists and belongs to B, and g is continuous at L

lim g(f(x)) = g lim f(x)

x -> a

ii) If f is continuous at a exists A and g is continuous at f(a) exists B, then g(f(x)) is continuous at a exists A

I is a non empty subset of R. A function f : I -> R is said to e bounded on I iff there is an M in R such that

|f(x)| < M

for all x exists I 

f is said to be dominated by M on I

I is closed and bounded

f : I -> R is continuous on I, then f is bounded on I, if:

M = sup f(x) and m = inf f(x)

then there exists points x_m, x_M exists I such that:

f(x_m) = m and f(x_M) = M

a < b and f : [a,b] -> R is continuous

If y₀ lies between f(a) and f(b), then there is an x₀ exists (a,b) such that f(x₀) = y₀

f : I -> R, for ever E > 0 there is d > 0 such that:

|x - a| < d and x,a exists I => |f(x) - f(a)| < E

I is a closed bounded interval

f : I -> R is continuous on I, then f is uniformly continuous on I

a < b and f : (a,b) -> R, then f is unifomly continuous on (a,b) iff f can be continuously extended to [a,b]; that is, iff there is a continuous function g : [a,b] -> R which satisfies

f(x) = g(x), x exists (a,b)

a exists R, iff f is defined on some open interval I containing a and

f'(a) := [lim f(a+h) - f(a)]/h

h -> 0

a exists R iff there exists and open interval I and a function F : I -> R such that a exists I, f is defined on I, F is continuous at a, and 

f(x) = F(x)(x - a) + f(a)

holds for all x in I, F(a) = f'(a)

f is diff. at a iff there is a function T of the form T(x) := mx such that

lim (f(a+h) - f(a) - T(h))/h = 0

h -> 0