# Answer to Question #15411 in Calculus for harry smith

Question #15411

(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.]

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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