Continuity and Differentiability — MATHS STD 12 Science — Question
Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
Discuss the continuity of the function: f(x) = sin x - cos x
✓
Answer
We known that if g and r are two continuous functions, then g + r, g – r and g.r are also continuous. First we have to prove that g(x) = sinx and r(x) = cosx are continuous functions. Now, let g(x) = sinx We know that g(x) = sinx is defined for every real number. Let h be a real number. Now, put x = h + k So, if x $\rightarrow$ h and k $\rightarrow$ 0 g(h) = sin h $\mathop {\lim }\limits_{x \to h} g(x) = \mathop {\lim }\limits_{x \to h} \sin x$ = $\mathop {\lim }\limits_{k \to 0} \sin (h + k)$ = $\mathop {\lim }\limits_{k \to 0} [\sinh \cos k + \cosh \sin k]$ = sinh.cos0 + cosh.sin0 = sinh + 0 = sin h Thus $\mathop {\lim }\limits_{x \to h} g(x) = g(h)$ Therefore, g is a continuous function …(1) Now, let f(x) = cos x We know that f(x) = cos x is defined for every real number. Let h be a real number. Now, put x = h + k So, if x $\rightarrow$ h and k $\rightarrow$ 0 Now f(h) = cosh $\mathop {\lim }\limits_{x \to h} {\text{f}}({\text{x}}) = \mathop {\lim }\limits_{{\text{x}} \to {\text{h}}} \cos {\text{x}}$ = $\mathop {\lim }\limits_{{\text{k}} \to 0} \cos ({\text{h}} + {\text{k}})$ = $\mathop {\lim }\limits_{x \to 0} [\cosh \cos k - \sinh \sin k]$ = coshcos0 - sinhsin0 = cosh - 0 = cosh Thus $\mathop {\lim }\limits_{x \to h} f(x) = f(h)$ Therefore, f is a continuous function ….(2) So, from (1) and (2), we get, r(x) = g(x) - f(x) = sinx - cosx is a continuous function.
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