Example 18 - Prove that f(x) = tan x is a continuous function - Algebra of continous functions

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Example 18 Prove that the function defined by f (x) = tan x is a continuous function. Let 𝑓﷐𝑥﷯ = tan⁡𝑥 𝑓﷐𝑥﷯ = ﷐﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯ is defined for all real number except cos⁡𝑥 = 0 i.e. x = ﷐2𝑛+1﷯ ﷐𝜋﷮2﷯ Let 𝑝﷐𝑥﷯=﷐sin﷮𝑥﷯ & 𝑞﷐𝑥﷯=﷐cos﷮𝑥﷯ We know that sin x & ﷐cos﷮𝑥﷯ is continuous for all real numbers. ⇒ 𝑝﷐𝑥﷯ & 𝑞﷐𝑥﷯ is continuous. By Algebra of continuous function If 𝑝﷐𝑥﷯ & 𝑞﷐𝑥﷯ both continuous for all real numbers then 𝑓﷐𝑥﷯= ﷐𝑝﷐𝑥﷯﷮𝑞﷐𝑥﷯﷯ , is continuous for all real numbers such that 𝑞﷐𝑥﷯ ≠ 0 ⇒ 𝑓﷐𝑥﷯ = ﷐﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯ is continuous for all real numbers such that ﷐cos﷮𝑥﷯ ≠ 0 i.e. 𝑥 ≠﷐2𝑛+1﷯﷐𝜋﷮2﷯ Hence, ﷐tan﷮𝑥﷯ is continuous at all real numbers except 𝒙=﷐𝟐𝒏+𝟏﷯﷐𝝅﷮𝟐﷯

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