up to 4-16

Question 1

One-sided to two sided limit equality

Answer 1

If the limit from the left and from the right at a certain point are the same then that limit is the limit at that point

Question 2

limit of f(x) goes to L when x goes to infinity iff

Answer 2

there exists a c >0 s.t. (c, inf) is in the domain of f, and given any e>0 there exists M >= c s.t. for all x > M we have |f(x) - L| < e

Question 3

limit of f(x) goes to L when x goes to -infinity iff

Answer 3

there exists a c >0 s.t (-inf, -c) is in the domain of f, and given any e > 0 there exists M <= -c s.t. for all x<M we have |f(x) - L| < e

Question 4

f diverges to infinity when x goes to a iff

Answer 4

given M>0 there exists d > 0 s.t. |x-a|<d where f(x)>M

Question 5

f diverges to -infinity when x goes to a iff

Answer 5

given M< 0 there exists d>0 s.t. |x-a|<d where f(x)<M

Question 6

Let E be a subset of the real numbers, f is continuous at point a in E iff

Answer 6

given any e>0 there exists d>0 s.t. for all x in E with |x-a|<d we have |f(x)-f(a)|<e

Question 7

Theorem of Sequential Continuity

Answer 7

Let E be a subset of the real numbers, f in E, f is continuous at E if lim(f(xn)) = f(a) for every sequence xn in E where lim xn = a

Question 8

let f:A->R and g:B->R, when A,B are subsets of the real numbers. If f(A) is a subset of B then we define the composition as:

Answer 8

the composition of g with f is g◦f : A->R given by
g ◦ f(x) = g(f(x))

Question 9

Composition and limits swapping

Answer 9

If lim f(x) = L as x->a is an element of B, and g is continuous at L, then lim g ◦ f(x) = g(lim f(x)) = g(L)

Question 10

Continuity and Composition

Answer 10

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

Question 11

Extreme Value Theorem

Answer 11

Let I be a CLOSED AND BOUNDED interval, with I=[a,b], and a,b both finite real numbers. Then
if f:I->R is continuous on I, then f is bounded on I
if M = sup( f(x) ) and m = inf( f(x) ), and both m, M are finite, then there exists x and y in I where f(x) = m and f(y) = M

Question 12

Intermediate Value Theorem

Answer 12

suppose f is continuous [a,b] on R. if y lies between f(a) and f(b) then there exists x in (a,b) s.t. f(x) = y