Question

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Solution

(i) Let f(x) = cos x. D_f = R
Let a be any real number ∈ D_f

$Now space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below cos space straight x space space space space space space space space space space space space space space space space space space space left square bracket Put space straight x equals straight a plus straight h space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a right square bracket space space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below cos left parenthesis straight a plus straight h right parenthesis equals Lt with straight a rightwards arrow 0 below left parenthesis cos space straight a space cos space straight h minus sin space straight a space sin space straight h right parenthesis space space space space space space space space space space space space space space space space space space space space space space equals cos space straight a space Lt with straight h rightwards arrow 0 below space cos space straight h minus sin space straight a space Lt with straight h rightwards arrow 0 below sin space straight h space space space space space space space space space space space space space space space space space space space space space space equals cos space straight a.1 minus sin space straight a.0 equals cos space straight a Also space space space space straight f left parenthesis straight a right parenthesis equals cos space straight a therefore space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis equals cos space straight a$
∴ f is continuous as x = a
But a is any real number ∈ D_f∴ f is continuous at every point of its domain.
i.e., cos x is continuous at every point of its domain.
$left parenthesis ii right parenthesis space Let space straight f left parenthesis straight x right parenthesis equals cosec space straight x equals fraction numerator 1 over denominator sin space straight x end fraction therefore space space straight D subscript straight f equals straight R minus open curly brackets straight x space colon space straight R semicolon sin space straight x equals 0 close curly brackets space space space space space space space space space space equals straight R minus open curly brackets straight x equals nπ semicolon straight n element of straight I close curly brackets Let space straight a space be space any space real space number element of straight D subscript straight f. space space space space space space space space space space space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below fraction numerator 1 over denominator sin space straight x end fraction space space space space space space space left square bracket Put space straight x equals straight a plus straight h space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator 1 over denominator sin left parenthesis straight a plus straight h right parenthesis end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator 1 over denominator sin space straight a space cos space straight h plus cos space straight a space sin space straight h end fraction space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator sin space straight a.1 plus cos space straight a.0 end fraction equals fraction numerator 1 over denominator sin space straight a end fraction Also space space space space space space space space space space space straight f left parenthesis straight a right parenthesis equals fraction numerator 1 over denominator sin space straight a end fraction therefore space space space space space space space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis$

∴ f is continuous at x = a
But a is any real number ∈ D_f∴ f is continuous function at every point of domain,
∴ cosec x is continuous at every point of domain.
$left parenthesis iii right parenthesis space Let space straight f left parenthesis straight x right parenthesis equals sec space straight x equals fraction numerator 1 over denominator cos space straight x end fraction space space space space space space straight D subscript straight f equals end subscript straight R space except space odd space multiples space of space straight pi over 2 Let space straight a space be space any space real space number space element of space space Df space space space space space space space space space space space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below fraction numerator 1 over denominator cos left parenthesis straight a plus straight h right parenthesis end fraction space space space space space left square bracket put space straight x equals space straight a plus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space cos space straight x plus straight a right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator 1 over denominator cos left parenthesis straight a plus straight h right parenthesis end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator 1 over denominator cos space straight a space cos space straight h minus sin space straight a space sin space straight h end fraction space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator cos space straight a.1 minus sin space straight a.0 end fraction equals fraction numerator 1 over denominator cos space straight a end fraction Also space space space space space space space space space space space straight f left parenthesis straight a right parenthesis equals fraction numerator 1 over denominator cos space straight a end fraction therefore Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis$
∴ f is continuous at x = a
But a is any real number ∈ D_f∴ f is continuous at every point of domain,
∴ sec x is continuous at every point of domain.
$left parenthesis iv right parenthesis space Let space straight f left parenthesis straight x right parenthesis equals cot space straight x equals fraction numerator cos space straight x over denominator sin space straight x end fraction space space space space space space space space space space space space straight D subscript straight f equals straight R minus open curly brackets straight x colon straight R semicolon sin space straight x equals 0 close curly brackets space space space space space space space space space space space space space space space space equals straight R minus open curly brackets straight x equals nπ comma straight n element of straight I close curly brackets Let space straight a space be space any space real space number space element of straight D subscript straight f space space space space space space space space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below fraction numerator cos space straight x over denominator sin space straight x end fraction space space space left square bracket Put space straight x equals straight a plus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a right square bracket space space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator cos left parenthesis straight a plus straight h right parenthesis over denominator sin left parenthesis straight a plus straight h right parenthesis end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator cos space straight a space cos space straight h minus sin space straight a space sin space straight h over denominator sin space straight a space cos space straight h plus cos space straight a space sin space straight h end fraction space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator cos space straight a.1 minus sin space straight a.0 over denominator sin space straight a.1 equals cos space straight a.0 end fraction equals fraction numerator cos space straight a over denominator sin space straight a end fraction Also space space space space space space space space straight f left parenthesis straight a right parenthesis equals fraction numerator cos space straight a over denominator sin space straight a end fraction therefore space space space space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis$
∴ f is continuous at x = a.
But a is any real number ∈ D_f.
∴ f is continuous at every point of domain,
∴ cot x is continuous at every point of domain.

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