# Answer to Question #15374 in Calculus for hsd

Question #15374

(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 E R such that m ≤ g(x) ≤ M for

all x E R (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.]

g(x) = (3 + sin (1/x-2)/ (1 + x^2)

is bounded. This means to find real numbers m, M E R such that m ≤ g(x) ≤ M for

all x E R (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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