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  •  Let [.] denote the greatest integer function and <img alt="\mathrm{f(x)=\left[\tan ^2 x\right]}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bf%28x%29%3D%5Cleft%5B%5Ctan%2

 Let [.] denote the greatest integer function and \mathrm{f(x)=\left[\tan ^2 x\right]}. Then,

Option: 1

\mathrm{\lim _{x \rightarrow 0} f(x)} does not exist


Option: 2

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


Option: 3

\mathrm{f\left ( x \right )} is not differentiable at \mathrm{x=0}


Option: 4

\mathrm{f^{\prime}(0)=1}.


Answers (1)

best_answer

We have \mathrm{f(0)=\left[\tan ^2 0\right]=[0]=0},

\mathrm{f(0+0) =\lim _{x \rightarrow 0} f(x)=\lim _{h \rightarrow 0}\left[\tan ^2(0+h)\right], \text { where } h \text { is positive and sufficiently small }}
                                         \mathrm{=\lim _{h \rightarrow 0}\left[\tan ^2 h\right]=\lim _{h \rightarrow 0} 0=0}

and \mathrm{ f(0-0)= \lim _{x \rightarrow 0-} f(x)=\lim _{h \rightarrow 0}\left[\tan ^2(0-h)\right], \text { where } h \text { is positive and sufficiently small }}
                          \mathrm{= \lim _{h \rightarrow 0}\left[\tan ^2 h\right]=\lim _{h \rightarrow 0} 0=0 }.


Since  f(0-0)=f(0)=f(0+9)$, therefore $f(x)$ is continuous at $x=0.
 

Posted by

Suraj Bhandari

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