Bound Function: Showing Continuity at All x ≠ 2 & x = 2

In summary, the conversation is discussing the function g(x) = [3 + sin(1/x-2)]/[1 + x^2] and its boundedness, as well as the function f(x) = { [x-2] [3 + sin(1/x-2)]/[1 + x^2] , if x ≠ 2, { 0 , if x = 2. The goal is to show that f(x) is continuous at all x ≠ 2 and at x = 2 using the Squeeze Theorem and considering the behaviors of u(x) and v(x) in g(x) = u(x)/v(x).
  • #1
hsd
6
0
(a) Show that the function g(x) =[3 + sin(1/x-2)]/[1 + x^2] is bounded.
This means to find real numbers m; M is an lR such that m ≤ g(x) ≤ M for
all x is an lR (and to show that these inequalities are satisfied!).

(b) Explain why the function:

f(x) = { [x-2] [3 + sin(1/x-2)]/[1 + x^2] , if x ≠ 2,
{ 0 , if x = 2.

is continuous at all x ≠ 2.

(c) Show that the function f(x) in Part (b) is continuous at x = 2. [Hint: Use
Part (a) and the Squeeze Theorem.]
 
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  • #2
how about considering g(x) = u(x)/v(x) and each of the behaviours of those functions

note that g(x) will get big whenever u(x)>>v(x)
 
  • #3
Welcome to PF;
How about showing us your attempt at the problem? ... that way we can target our assistance to where you need it most.
 

FAQ: Bound Function: Showing Continuity at All x ≠ 2 & x = 2

What is a bound function?

A bound function is a mathematical function that has a defined range within a specific domain. This means that the output of the function is limited to a certain set of values.

How do you determine continuity at all x ≠ 2 & x = 2?

To determine continuity at all x ≠ 2 & x = 2, we must first check if the function is defined at x = 2. If it is, we then evaluate the limit as x approaches 2 from both the left and right sides. If the limit exists and is equal from both sides, the function is continuous at x = 2. We also need to ensure that the function is continuous at all other values of x.

What does it mean for a function to be continuous?

A function is continuous if there are no breaks or jumps in the graph of the function. This means that the function is defined at all points in its domain and the limit of the function exists at all points. In other words, the graph of the function is a continuous curve with no gaps or holes.

Why is continuity important in mathematics?

Continuity is important in mathematics because it allows us to make accurate predictions and interpretations about the behavior of a function. It also helps us to understand the fundamental properties of a function and how it relates to other mathematical concepts. Continuity is also essential in calculus, as it allows us to use derivative and integral rules to solve problems involving continuous functions.

How can we prove the continuity of a function at a specific point?

To prove the continuity of a function at a specific point, we must show that the function is defined at that point, the limit of the function exists at that point, and the limit is equal to the value of the function at that point. This can be done using algebraic manipulation, the definition of a limit, or using theorems and properties of continuous functions.

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