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***Please Avoid This Question**** $$\lim_{x\rightarrow 0}\frac{27^x - 9^x - 3^x + 1}{\sqrt{2} - \sqrt{1+\cos x}}$$

I tried to solve it by applying the L'Hospital's rule.

$$\lim_{x\rightarrow 0}\frac{27^x\ln27- 9^x\ln9- 3^x\ln3 + 0}{(-1/2)/\sqrt{1+\cos x}}$$

Now simply apply the limit,

$$\lim_{x\rightarrow 0}\frac{1\ln27- 1\ln9 - 1\ln3}{(-1/2)/\sqrt{2}}=0$$

But the correct answer in my textbook is $8\sqrt{2}(\ln3)^2$

Where am I doing incorrect?

Edit: I did a silly mistake while Solving it.

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  • $\begingroup$ rationalise the denominator. now write the 1- cos x as $2 sin^2 x$ $\endgroup$ – maveric Feb 15 at 15:55
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The derivative of $\sqrt{1+\cos x}$ is $\dfrac{-\sin x}{2\sqrt{1+\cos x}}$

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  • $\begingroup$ Oh! I did a Silly Mistake. $\endgroup$ – Navneet Kumar Feb 15 at 15:50

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