Question

Can anyone help me out with this question​

Answer 1

Step-by-step explanation:

We have,

$\lim_{x \rarr0} \frac{ {27}^{x} - {9}^{x} - {3}^{x} + 1 } { \sqrt{2} - \sqrt{1 + \cos(x) } }$

This is in $\frac{0}{0}$ form, so using l'hospital rule

$\lim_{x \rarr0} \frac{ {27}^{x}. ln(27) - {9}^{x} . ln(9) - {3}^{x} . ln(3) }{ \frac{ \sin(x) }{2 \sqrt{1 + \cos(x) } } }$

$= \lim_{x \rarr0} \frac{2( {27}^{x}. ln(27) - {9}^{x} . ln(9) - {3}^{x} . ln(3) ) }{ \sqrt{1 - \cos(x) } }$

$= \lim_{x \rarr0} 2( {27}^{x}. ln(27) - {9}^{x} . ln(9) - {3}^{x} . ln(3) ) . \lim_{x \rarr0} \frac{ \sqrt{1 + \cos(x) } }{ \sin(x) }$

$= 2( ln(27) - ln(9) - ln(3) ) . \lim_{x \rarr0} \frac{ \sqrt{1 + \cos(x) } }{ \sin(x) }$

$= 0$

Answer 2

Answer:

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