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question no.8 

​​ question no.8  x + xo+xq—xuyse 6t turr (zx€)tgs •O = x (x)JJ0 OS (0)/ JO omen ptl!d x um.x

Dear student,

$$\begin{gathered} \underset{\mathrm{x}\to 0}{\lim } \frac{\sin 3{\mathrm{x}}^2}{\ln \left[\cos \left(2{\mathrm{x}}^2-\mathrm{x}\right)\right]} \\ =\underset{\mathrm{x}\to 0}{\lim }\frac{6\mathrm{x} . \cos 3{\mathrm{x}}^2}{{\displaystyle \frac{1}{\cos \left(2{\mathrm{x}}^2-\mathrm{x}\right)}} \times \left[- \sin \left(2{\mathrm{x}}^2-\mathrm{x}\right)\right]\times \left(4\mathrm{x} - 1\right)} \left[\mathrm{Using} \mathrm{L} \mathrm{Hospital}\text{'}\mathrm{s} \mathrm{rule}\right] \\ =\underset{\mathrm{x}\to 0}{\lim } \frac{\left(-6\right) . \cos 3{\mathrm{x}}^2 .\cos \left(2{\mathrm{x}}^2-\mathrm{x}\right)}{\left(4\mathrm{x} - 1\right)} \times \frac{\mathrm{x}}{\sin \left(2{\mathrm{x}}^2-\mathrm{x}\right)} \\ =-6\underset{\mathrm{x}\to 0}{\lim } \frac{\cos 3{\mathrm{x}}^2 .\cos \left(2{\mathrm{x}}^2-\mathrm{x}\right)}{\left(4\mathrm{x} - 1\right)} \times \mathrm{x} \times \frac{1}{{\displaystyle \frac{\sin \left(2{\mathrm{x}}^2-\mathrm{x}\right)}{2{\mathrm{x}}^2-\mathrm{x}}} \times \left(2{\mathrm{x}}^2-\mathrm{x}\right)} \\ =-6\underset{\mathrm{x}\to 0}{\lim } \frac{\cos 3{\mathrm{x}}^2 .\cos \left(2{\mathrm{x}}^2-\mathrm{x}\right)}{\left(4\mathrm{x} - 1\right)} \times \underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}}{\mathrm{x}\left(2\mathrm{x} - 1\right)} \times \underset{\mathrm{x}\to 0}{\lim }\frac{1}{{\displaystyle \frac{\sin \left(2{\mathrm{x}}^2-\mathrm{x}\right)}{ \left(2{\mathrm{x}}^2-\mathrm{x}\right)}}} \\ =-6\times \frac{ \cos 0 . \cos 0}{\left(4\times 0-1\right)} \times \frac{1}{\left(2\times 0-1\right)} \times \frac{1}{1} \left[\mathrm{as}, \underset{\mathrm{\theta }\to 0}{\lim }\frac{\sin \mathrm{\theta }}{\mathrm{\theta }} = 1\right] \\ =-6 \times \frac{1 \times 1}{\left(-1\right)}\times \frac{1}{\left(-1\right)} \\ =-6 \end{gathered}$$
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