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Example 18 - Prove that f(x) = tan x is a continuous function

Example 18 - Chapter 5 Class 12 Continuity and Differentiability - Part 2


Example 18 Prove that the function defined by f (x) = tan x is a continuous function.Let 𝑓(π‘₯) = tan⁑π‘₯ 𝒇(𝒙) = 𝐬𝐒𝐧⁑𝒙/πœπ¨π¬β‘π’™ Here, 𝑓(π‘₯) is defined for all real number except 𝒄𝒐𝒔⁑𝒙 = 0 i.e. for all x except x = (πŸπ’+𝟏) 𝝅/𝟐 Let 𝑝(π‘₯)=sin⁑π‘₯ & π‘ž(π‘₯)=cos⁑π‘₯ We know that sin x & cos⁑π‘₯ is continuous for all real numbers. Therefore, 𝑝(π‘₯) & π‘ž(π‘₯) is continuous. By Algebra of continuous function If 𝑝, π‘ž are continuous , then 𝒑/𝒒 is continuous. Thus, Rational Function 𝑓(π‘₯) = sin⁑π‘₯/cos⁑π‘₯ is continuous for all real numbers except at points where π‘π‘œπ‘  π‘₯ = 0 i.e. π‘₯ β‰ (2𝑛+1) πœ‹/2 Hence, tan⁑π‘₯ is continuous at all real numbers except 𝒙=(πŸπ’+𝟏) 𝝅/𝟐

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.