# Find limit “1 ^ infinity”

$\displaystyle \lim_\limits{x \to \frac{\pi}{4}} \left[\frac{1 + \sin(\pi/4 - x)\sin(2x)}{1 + 0.5\cos(2x)}\right]^{\cot^3(x - \frac{\pi}{4})}$

I've tried to use $\exp$, but found very difficult to simplify the expression.

• Just curious, is the use of rounding brackets [ ] intentional? Or did you mean to use parens ( ) -- en.wikipedia.org/wiki/Nearest_integer_function – Albert Renshaw Oct 25 '15 at 22:05
• Do you want thorough precision and "rigor" in finding the limit, or would you be content with asymptotic relationships (that are ultimately just as rigorous if you take the time to prove their validity)? – alex.jordan Oct 25 '15 at 22:05
• @AlbertRenshaw: It's common to use brackets for additional grouping, especially in a problem like this where there are already many parenthesis flying around. Unless someone specifically says that $[\cdot]$ denotes the nearest integer function (or floor function, etc.), it's safe to assume that they're used solely for grouping. – anomaly Oct 25 '15 at 22:09
• @anomaly thanks! – Albert Renshaw Oct 25 '15 at 22:18

$$\displaystyle \lim_\limits{x \to \frac{\pi}{4}} \left(\frac{1 + \sin\left(\frac{\pi}{4} - x\right)\sin(2x)}{1 + \frac{1}{2}\cos(2x)}\right)^{\cot^3(x - \frac{\pi}{4})}=$$

$$\displaystyle \lim_\limits{x \to \frac{\pi}{4}} \left(\frac{2+\sqrt{2}(\cos(x)-\sin(x))\sin(2x)}{2+\cos(2x)}\right)^{-\tan^3\left(\frac{\pi}{4}+x\right)}=$$

$$\lim_{x \to \frac{\pi}{4}} \exp\left(\ln\left(\left(\frac{2+\sqrt{2}(\cos(x)-\sin(x))\sin(2x)}{2+\cos(2x)}\right)^{-\tan^3\left(\frac{\pi}{4}+x\right)}\right)\right)=$$

$$\lim_{x \to \frac{\pi}{4}} \exp\left(-\ln\left(\frac{2+\sqrt{2}(\cos(x)-\sin(x))\sin(2x)}{2+\cos(2x)}\right)\tan^3\left(\frac{\pi}{4}+x\right)\right)=$$

$$\exp\left(\lim_{x \to \frac{\pi}{4}} -\ln\left(\frac{2+\sqrt{2}(\cos(x)-\sin(x))\sin(2x)}{2+\cos(2x)}\right)\tan^3\left(\frac{\pi}{4}+x\right)\right)=$$

$$\exp\left( -\left(\lim_{x \to \frac{\pi}{4}}\ln\left(\frac{2+\sqrt{2}(\cos(x)-\sin(x))\sin(2x)}{2+\cos(2x)}\right)\tan^3\left(\frac{\pi}{4}+x\right)\right)\right)=$$

$$\exp\left( -\left(\lim_{x \to \frac{\pi}{4}}\sin^3\left(x+\frac{\pi}{4}\right)\csc^3\left(\frac{\pi}{4}-x\right)\ln\left(\frac{\sqrt{2}\sin(2x)(\cos(x)-\sin(x))+2}{2+\cos(2x)}\right)\right)\right)=$$

$$\exp\left( -\left(\lim_{x \to \frac{\pi}{4}}\csc^3\left(\frac{\pi}{4}-x\right)\ln\left(\frac{\sqrt{2}\sin(2x)(\cos(x)-\sin(x))+2}{2+\cos(2x)}\right)\right)\right)=$$

$$\exp\left(-\left(-\frac{3}{2}\right)\right)=e^{--\frac{3}{2}}=e^{\frac{3}{2}}$$