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Continuity and Differentiability

Short Answer Type

71. Examine the following functions for continuity :

straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator straight x minus 5 end fraction

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72. Examine the following functions for continuity :

straight f left parenthesis straight x right parenthesis equals fraction numerator straight x squared minus 25 over denominator straight x minus 5 end fraction

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73. Examine the following functions for continuity :

f(x) = | x – 5 |

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74. Discuss the continuity of sine function

Let f(x) = sin x D_f = R
Let a be any real number ∈ D_f

$Now space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below sin space straight x space space space space space space space space space space space space space space space space space space space space space left square bracket Put space straight x equals straight a plus straight h space so space taht space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a right square bracket space space space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below sin left parenthesis straight a plus straight h right parenthesis space space space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below left parenthesis sin space straight a space cos space straight h plus space cos space straight a space sin space straight h right parenthesis space space space space space space space space space space space space space space space space space space space space space space space equals sin space straight a Lt with straight h rightwards arrow 0 below cos space straight h plus cos space straight a space Lt with straight h rightwards arrow 0 below sin space straight h space space space space space space space space space space space space space space space space space space space space space space space equals sin space straight a.1 plus cos space straight a.0 equals sin space straight a Also space straight f left parenthesis straight a right parenthesis equals space sin space straight a therefore space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis equals sin space straight a$
∴ f is continuous at x = a
But a is any real number ∈ D_f∴ f is continuous function
∴ sin x is continuous at every point of its domain,