Question

Examine the following functions for continuity :
$straight f left parenthesis straight x right parenthesis equals fraction numerator straight x squared minus 25 over denominator straight x minus 5 end fraction$

Solution

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

For f to be defined,
x + 5 ≠ 0, i.e. x ≠ – 5
∴D_f = Set of all real numbers except – 5 = R - {- 5}
Let c≠– 5 be any real number.

therefore space straight f left parenthesis straight c right parenthesis equals fraction numerator straight c squared minus 25 over denominator straight c plus 25 end fraction equals fraction numerator left parenthesis straight c minus 5 right parenthesis left parenthesis straight c plus 5 right parenthesis over denominator straight c plus 5 end fraction equals straight c minus 5 Also space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below fraction numerator straight x squared minus 25 over denominator straight x plus 5 end fraction equals Lt with straight x rightwards arrow straight c below fraction numerator left parenthesis straight x minus 5 right parenthesis left parenthesis straight x plus 5 right parenthesis over denominator straight x plus 5 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 equals Lt with straight x rightwards arrow straight c below left parenthesis straight x minus 5 right parenthesis equals straight c minus 5 therefore space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at x = c.
But c ≠ – 5 is any real number.
∴ f is continuous at every real number c ∈ D_f.
∴ f is continuous function.

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