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  • A function   is defined as below <img alt="\mathrm{ f(x)=\frac{\cos (\si

A function \mathrm{f(x)}  is defined as below

\mathrm{ f(x)=\frac{\cos (\sin x)-\cos x}{x^2}, x \neq 0} and  \mathrm{f(0)=a }.

\mathrm{f(x) } is continuous at \mathrm{x=0}  if a equals

Option: 1

0


Option: 2

4


Option: 3

5


Option: 4

6


Answers (1)

best_answer

\begin{aligned} \text{RH limit }&=\mathrm{\lim _{h \rightarrow 0} \frac{\cos \{\sin (0+h)\}-\cos (0+h)}{(0+h)^2}=\lim _{h \rightarrow 0} \frac{\cos \sin h-\cos h}{h^2} }\\ & =\mathrm{\lim _{h \rightarrow 0} \frac{-\sin (\sin h) \times \cos h+\sin h}{2 h}} \\ & =\mathrm{\lim _{h \rightarrow 0} \frac{-\cos (\sin h) \times \cos ^2 h+\sin (\sin h) \times \sin h+\cos h}{2}=0 .}\\ \end{aligned}\\ \text{Similarly, LH limit =0.}\\ \text{As f(x) is continuous at x=0, f(0)=0.}

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