Ex 5.1

Chapter 5 Class 12 Continuity and Differentiability
Serial order wise

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### Transcript

Ex 5.1, 22 (i) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Let π(π)=ππ¨π¬β‘π To check continuity of π(π₯), We check itβs if it is continuous at any point x = c Let c be any real number f is continuous at π₯ =π if if L.H.L = R.H.L = π(π) i.e. limβ¬(xβπ^β ) π(π₯)= limβ¬(xβπ^+ ) " " π(π₯)= π(π) LHL at x β c limβ¬(xβπ^β ) f(x) = limβ¬(hβ0) f(c β h) = limβ¬(hβ0) cosβ‘(πββ) = limβ¬(hβ0) sinβ‘π sinβ‘β+cosβ‘π cosβ‘β Putting β=0 = sinβ‘π sinβ‘0+cosβ‘π cosβ‘0 = 0+cos c . 1 = πππβ‘π π΄π , cosβ‘(π₯βπ¦) =cosβ‘π₯ cosβ‘π¦βsinβ‘π₯ sinβ‘π¦ RHL at x β c limβ¬(xβπ^+ ) f(x) = limβ¬(hβ0) f(c + h) = limβ¬(hβ0) cosβ‘(π+β) = limβ¬(hβ0) cosβ‘π cosβ‘β β sinβ‘π sinβ‘β Putting β=0 = cosβ‘π cosβ‘0 β sinβ‘π sinβ‘0 = cos c . 1 β 0 = πππβ‘π π΄π , cosβ‘(π₯+π¦) =cosβ‘π₯ cosβ‘π¦βsinβ‘π₯ sinβ‘π¦ And, π(π) = cosβ‘π Since, L.H.L = R.H.L = π(π) β΄ Function is continuous at x = c Thus, we can write that f is continuous for x = c , where c βπ β΄ πππβ‘π is continuous for every real number.