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  4. The value of displaystyle limx→2 (e3x-6 -1/ sin (2-x)) is

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Q. The value of $ \displaystyle\lim_{x\to2} \frac{e^{3x-6} -1}{\sin \left(2-x\right)} $ is

WBJEEWBJEE 2008Limits and Derivatives

Solution:

$\displaystyle\lim_{x\to2} \frac{e^{3x-6} -1}{\sin \left(2-x\right)} $

$=\displaystyle \lim_{x\to 2} -\frac{ e^{3x-6}\left(3\right)}{\cos \left(2-x\right)}$

(using L’Hospital’s rule)

$= -\frac{3e^{0}}{\cos\, 0} = -3$