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Question
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
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Solution
(i) f(x) = cos (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (cos c) = (cos c) = (cos c)
This statement is true; that is, f(x) is continuous at every point on the real number line.
(ii) f(x) = cosec (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x ->^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (cosec c) = (cosec c) = (cosec c)
This statement is true; that is, f(x) is continuous at every point on the real number line.
(iii) f(x) = sec (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (sec c) = (sec c) = (sec c)
This statement is true; that is, f(x) is continuous at every point on the real number line.
(iv) f(x) = cot (x)
Let c be any real number such that (n − 1)π < x < nπ, where n represents an integer point.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (cot c) = (cot c) = (cot c)
This statement is true; that is, f(x) is continuous at every point on the real number line between (n − 1)π and nπ.
Now if we consider c such that c = nπ, where n represents an integer point, then:
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ ±∞ = ±∞ = ±∞
That is, f(x) is continuous at every point on the real number line except at the nπ-type points.
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