For b ∈ℝ, consider the function gb:(0, ∞) →(0, ∞) defined by gb(x)= x^b . Show that gb1 ∘gb2=gb1

For b ∈ℝ, consider the function gb:(0, ∞) →(0, ∞) defined by...

Key Concepts

Function Composition

Function composition involves applying one function to the result of another. In this context, composing two functions means substituting the output of one power function into the input of another, effectively combining their actions into a single function transformation. This concept is central when analyzing how successive operations interact with one another.

Power Functions

Power functions are functions of the form f(x) = x raised to a constant exponent. They are fundamental in mathematical analysis due to their straightforward behavior and the rich structure they display, which is useful for modeling various natural and theoretical processes, especially when the domain is restricted to positive real numbers.

Properties of Exponents

The rule (x^a)^b = x^(ab) is a fundamental property of exponents that allows for the simplification of nested power expressions. This property is essential in proving how the composition of power functions results in a function that can be represented by a single exponent, namely the product of the individual exponents.

Commutativity of Exponent Multiplication

Since multiplication of real numbers is commutative, the product of two exponents does not depend on their order. This property implies that composing two power functions yields the same result regardless of the order of composition, thereby ensuring that the resulting function is invariant under the interchange of the two component functions.

Transcript

00:01 Okay, what we want to do is we want to walk through a proof, and what we want to prove is that if we are told that the limit of f of x as x goes to infinity is equal to some numerical value, and the limit as x goes to infinity of g of x is equal to another numerical value, then the limit as x goes to infinity of f of x plus g of x has to equal a plus b.

00:26 And so this proof is going to be very similarly similar to ones that maybe you have done in a previous section based off of that epsilon delta definition of a limit.

00:39 And we're going to actually use kind of that same scenario of a proof in combination with the definition of limits as x goes to positive infinity.

00:56 And so what we're going to do is we're going to let, you know, epsilon, of course, be greater than zero.

01:07 And since we know, and we're going to stick with this first limit first, since we know that the limit as x goes to infinity of f of x is equal to a, then, of course, there is a.

01:31 Corresponding number and and typically we would say capital m but we're going to have to define them since we have two limits we're going to do that as m1 such that x is greater than m1 is going to imply that f of x minus a is equal to epsilon over two and of course we're going to do epsilon over two and not epsilon because we have combining multiple limits.

02:07 Okay.

02:08 So this is just kind of like that definition of limits as x goes to positive infinity.

02:15 And then we're going to similarly do the same thing for the limit as g of x as well.

02:21 So this one we're going to let epsilon be greater than zero.

02:27 And since the limit as x goes to infinity of g of x is equal to b, then there is a number, a corresponding number, excuse me.

02:47 Because we need to keep consistent...

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