Mathematics - Area Under Curves Question with Solution | TestHub

Check the continuity of the function $\begin{array}{l}\mathrm{f}(\mathrm{x})=\left[{\tan }^2\mathrm{x}\right]\text{ at }\mathrm{x}=0.\\ \text{ L.H.L. (at }\mathrm{x}=0\text{ ) }\end{array}$ $\begin{array}{l}={\lim }_{x\to 0^{-}}\left[{\tan }^{2}x\right]={\lim }_{h\to 0}\left[{\tan }^2(0-h)\right]\\ ={\lim }_{h\to 0}\left[{\tan }^{2}h\right]=\left[{\tan }^{2}0\right]=[0]=0\\ \text{ R.H.L. }(\mathrm{at}x=0)\\ ={\lim }_{x\to 0^{+}}\left[{\tan }^{2}x\right]={\lim }_{h\to 0}\left[{\tan }^2(0+h)\right]\end{array}$ $={\lim }_{h\to 0}\left[{\tan }^{2}h\right]=\left[{\tan }^{2}0\right]=[0]=0$ Now, determine the value of $f(x)$ at $x=0$. $\mathrm{f}(0)=\left[{\tan }^{2}0\right]=[0]=0$ Hence, $f(x)$ is continuous at $x=0$.