lim-x-pi-2-cos-2-x-2-2-1-cos-x-x-pi-3-

Question Number 19271 by Joel577 last updated on 08/Aug/17

\lim_{x \to π} {(2 - \cos^{2} x)}^{\frac{2 \sqrt{2 (1 + \cos x)}}{{(x - π)}^{3}}}

Answered by ajfour last updated on 09/Aug/17

= \lim_{x \to π} {{[1 + \sin^{2} (π - x)]}^{\frac{1}{\sin^{2} (π - x)}}}^{4 \cos (\frac{x}{2}) \times \frac{\sin^{2} (π - x)}{{(π - x)}^{2}} \times \frac{- 1}{(π - x)}}

= e^{\lim_{x \to π} ∣ 4 \sin (\frac{π}{2} - \frac{x}{2}) ∣ \times \frac{\sin^{2} (π - x)}{{(π - x)}^{2}} \times \frac{- 1}{(π - x)}}

= e^{- \lim_{x \to π} \frac{∣ 4 \sin (\frac{π}{2} - \frac{x}{2}) ∣}{2 (\frac{π}{2} - \frac{x}{2})}}

R . H . L . = e^{2}; L . H . L . = e^{- 2} = \frac{1}{e^{2}} .

so the limit dont exist at x = π .

Commented by Joel577 last updated on 09/Aug/17

Can u explain with more detail Sir ?

I dont understand from the second row

Commented by ajfour last updated on 09/Aug/17

since \lim_{x \to 0} {(1 + x)}^{1 / x} = e

here we have \lim_{x \to π} {[1 + \sin^{2} (π - x)]}^{\frac{1}{\sin^{2} (π - x)}} = e

. .

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