Prove the following: Let f and g be real-valued functions defined on (a, infinity). Suppose that...

Prove the following:

Let f and g be real-valued functions defined on (a, infinity). Suppose that lim{x to infinity} f(x) = L and lim{x to infinity} g(x) = M, where L and M are real. Then lim{x to infinity} (fg)(x) = LM.

You must use the following definition: L is the limit of f, and we write that lim{x to infinity} f(x) = L provided that for each epsilon > 0 there exists a real number N > a such that x > N implies that |f(x)-L| < epsilon.

Expert Solution

Colby Messinger answered 2 hours ago

1.29 Prove or disprove that this is a vector space: the real-valued functions f of one...

1.29 Prove or disprove that this is a vector space: the real-valued functions f of one real variable such that f(7) = 0.

Let f : N → N and g : N → N be the functions defined...

Let f : N → N and g : N → N be the functions defined as ∀k ∈ N f(k) = 2k and g(k) = (k/2 if k is even, (k + 1) /2 if k is odd). (1) Are the functions f and g injective? surjective? bijective? Justify your answers. (2) Give the expressions of the functions g ◦ f and f ◦ g? (3) Are the functions g ◦ f and f ◦ g injective? surjective? bijective?...

Let F and G~be two vector fields in R2 . Prove that if F~ and G~...

Let F and G~be two vector fields in R2 . Prove that if F~ and G~ are both conservative, then F~ +G~ is also conservative. Note: Give a mathematical proof, not just an example.

Prove 1. Let f : A→ B and g : B → C . If g...

Prove 1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f is one-to-one. 2. Equivalence of sets is an equivalence relation (you may use other theorems without stating them for this one).

Which of the following are subspaces of the vector space of real-valued functions of a real...

Which of the following are subspaces of the vector space of real-valued functions of a real variables? (must select all of the subspaces.) A. The set of even function (f(-x) = f(x) for all numbers x). B. The set of odd functions (f(-x) = -f(x) for all real numbers x). C. The set of functions f such that f(0) = 7 D. The set of functions f such that f(7) = 0

Let f: A ->B and g:B -> A be functions. Prove that if fog is one-to-one...

Let f: A ->B and g:B -> A be functions. Prove that if fog is one-to-one and gof is onto, then f is a bijection.

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information...

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a...

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore,...

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore, assume that ⇀∇×⇀F=⇀∇×⇀G.Show that there is a scalar function f such that ⇀G=⇀F+⇀∇f.

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto function...

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto function Let f: X→Y and g: Y→Z be both onto. Prove that f◦g is an onto function Let f: X→Y and g: Y→Z be both one to one. Prove that g◦f is an one to one function Let f: X→Y and g: Y→Z be both one to one. Prove that f◦g is an one to one function

Question