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 know that if g and k are two continuous functions, then, g + k, g – k and g.k are also continuous. First we have to prove that g(x) = sin x and k(x) = cos x 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) = sinh $\mathop {\lim }\limits_{x \to h} g(x) = \mathop {\lim }\limits_{x \to h} \sin x$ = $\mathop {\lim }\limits_{x \to 0} \sin (h + k)$ = $\mathop {\lim }\limits_{x \to 0} [\sinh \cos k + \cosh \sin k]$ = sin h.cos 0 + cos h.sin 0 = sinh + 0 = sin h Thus, $\mathop {\lim }\limits_{x \to h} g(x) = g(h)$ Therefore, g is a continuous function ...(1) Now, let k(x) = cos x We know that k(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 k(h) = cosh $\mathop {\lim }\limits_{x \to h} {\text{k}}({\text{x}}) = \mathop {\lim }\limits_{{\text{x}} \to {\text{h}}} \cos {\text{x}}$ = $\mathop {\lim }\limits_{{\text{k}} \to 0} \cos ({\text{h}} + {\text{k}})$ = $\mathop {\lim }\limits_{x \to 0} [\cosh \cos k - \sinh \sin k]$ = cos h.cos 0 - sin h.sin 0 = cos h - 0 = cos h Thus, $\mathop {\lim }\limits_{x \to h} k(x) = k(h)$ Therefore, k is a continuous function ...(2) So, from (1) and (2), we get, f(x) = g(x) + k(x) = sinx + cosx is a continuous function.
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