Calculate $\lim_{x\to \pi/4}\cot(x)^{\cot(4*x)}$ without L'Hôpital's rule

Question

How can I calculate limit $$\lim_{x\to \pi/4}\cot(x)^{\cot(4*x)}$$ without using L'Hôpital's rule?

What I have tried so far:

I tried to use the fact that $\lim_{\alpha\to 0}(1 + \alpha)^{1/\alpha} = e$ and do the following: $$\lim_{x\to \pi/4}\cot(x)^{\cot(4 \cdot x)} = \lim_{x\to \pi/4}(1 + (\cot(x) - 1))^{\cot(4 \cdot x)} = \lim_{x\to \pi/4}(1 + (\cot(x) - 1))^{\frac{1} {\cot(x) - 1} \cdot (\cot(x) - 1) \cdot \cot(4 \cdot x)} = \lim_{x\to \pi/4}e^{(\cot(x) - 1) \cdot \cot(4 \cdot x)} = e^{\lim_{x\to \pi/4}{(\cot(x) - 1) \cdot \cot(4 \cdot x)}} $$ But I have problems calculating limit $$\lim_{x\to \pi/4}{(\cot(x) - 1) \cdot \cot(4 \cdot x)}$$ I tried to turn $\cot(x)$ into $\frac{\cos(x)}{\sin(x)}$ as well as turning it into $\tan(x)$, but I do not see any workaround afterwards.

I would appreciate any pieces of advice. Thank you!

Answer 1

$$\lim_{x\to\pi/4}\cot x^{\cot4x}=\left(\lim_{x\to\pi/4}(1+\cot x-1)^{1/(\cot x-1)}\right)^{\lim_{x\to\pi/4}{\cot4x(\cot x-1)}}$$

The inner limit converges to $e$

For the exponent, $$\lim_{x\to\pi/4}\cot4x(\cot x-1)=\lim_{x\to\pi/4}\dfrac{\cos4x}{\sin x}\cdot\lim_{x\to\pi/4}\dfrac{\cos x-\sin x}{\sin4x}$$

Now

$$\lim_{x\to\pi/4}\dfrac{\cos4x}{\sin x}=\dfrac{\cos\pi}{\sin\dfrac\pi4}=?$$

Finally

Method$\#:1$

$$F=\lim_{x\to\pi/4}\dfrac{\cos x-\sin x}{\sin4x}=\lim_{x\to\pi/4}\cos x\cdot\lim_{x\to\pi/4}\dfrac{\cot x- 1}{\sin4x}=\dfrac1{\sqrt2}\cdot\lim_{x\to\pi/4}\dfrac{\cot x- \cot\dfrac\pi4}{\sin4x-\sin\pi}$$

Method$\#:1A$

$$F=\dfrac1{\sqrt2}\cdot\dfrac{\dfrac{d(\cot x)}{dx}}{\dfrac{d(\sin4x)}{dx}}_{\text{at } x=\pi/4}$$

Method$\#:1B$

$$F=\dfrac1{\sqrt2}\cdot\lim_{x\to\pi/4}\dfrac1{\sin x\sin\dfrac\pi4} \cdot\lim_{x\to\pi/4}\dfrac{\sin\left(\dfrac\pi4-x\right)}{\sin4\left(\dfrac\pi4-x\right)}=\dfrac{\sqrt2}4$$

Method$\#:2$

set $\dfrac\pi4-x=y$ $$F=\sqrt2\lim_{x\to\pi/4}\dfrac{\sin\left(\dfrac\pi4-x\right)}{\sin4x}=\sqrt2\lim_{y\to0}\dfrac{\sin y}{\sin4\left(\dfrac\pi4-y\right)}=\dfrac{\sqrt2}4$$

Answer 2

With a subtitution $u = x-\frac{\pi}4$ you get

\begin{eqnarray}\lim_{u\to 0} \left[\frac{\cos\left(u+\frac{\pi}4\right)}{\sin\left(u+\frac{\pi}4\right)} -1\right]\cot(4u+\pi).\end{eqnarray}

Using periodicity and sum of angles you now have

$$\lim_{u\to 0} \left[\frac{\frac{\sqrt{2}}2\cos u - \frac{\sqrt 2}{2}\sin u}{\frac{\sqrt{2}}2\cos u + \frac{\sqrt 2}{2}\sin u} -1\right]\cot(4u).$$

Least common denominator brings you to

$$\lim_{u\to 0} \frac{-\sqrt 2 \sin u}{\frac{\sqrt{2}}2\cos u + \frac{\sqrt 2}{2}\sin u}\cdot \frac{\cos 4u}{\sin 4u}=\frac{-\sqrt 2}{\frac{\sqrt 2}2}\cdot \frac14=-\frac12.$$

Thus your limit is $\frac1{\sqrt e}.$

Answer 3

You have $\cot(x+y) = \frac{\cot x \cot y -1}{\cot x + \cot y}$

Taking $h = x-\pi/4$ you get

$$\cot x = \cot(\pi/4 + h) = \frac{\cot h -1}{\cot h + 1} = \frac{\cos h - \sin h}{\cos h + \sin h} = 1- 2h +o(h)$$ around zero.

And with $4x = \pi + 4h$

$$\cot(4x) = \cot(\pi + 4h) = \frac{\cos(\pi+4h)}{\sin(\pi+4h)} = \frac{\cos(4h)}{\sin(4h)} = \frac{1}{4h} +o\left(\frac{1}{h}\right)$$

Now $$\lim\limits_{h \to 0} (1-2h)^{\frac{1}{4h}} = 1/\sqrt{e}$$ which is the limit you're looking for.

Answer 4

I would use the ansatz $$\lim_{x\to \frac{\pi}{4}}\exp(\cot(4x)\log(\cot(x)))$$ $$=\exp\left(\lim_{x\to \frac{\pi}{4}}\frac{\cos(4x)\log(\cot(x))}{\sin(4x))}\right)$$ $$=\exp\left(\lim_{x\to \frac{\pi}{4}}\cos(4x)\lim_{x\to \frac{\pi}{4}}\frac{\log(\cot(x))}{\sin(4x)}\right)$$ $$=\exp\left(-1\lim_{x\to \frac{\pi}{4}}\frac{\log(\cot(4x))}{\sin(4x)}\right)$$

Answer 5

We substitute $c = \cot(x)$ and we get $$ \lim\limits_{x\to\pi/4} \cot(x)^{\cot(4x)} =\lim\limits_{c\to 1}\; c^{\frac{c^4-6c^2+1}{4c^3-4c}} $$ because $$ \cot(4x) = \frac{\cot^4x-6\cot^2x+1}{4\cot^3x-4\cot x} $$ Then we have $$ \lim\limits_{c\to 1} \;c^{\frac{c^4-6c^2+1}{4c^3-4c}} =\lim\limits_{c\to 1} \;\exp\left(\ln(c) \cdot \frac{c^4-6c^2+1}{4c^3-4c}\right) =\lim\limits_{c\to 1} \;\exp\left(\frac{\ln(c)}{c-1} \cdot \frac{c^4-6c^2+1}{4c^2+4c}\right) \\ =\exp\left(\lim\limits_{c\to 1} \;\frac{\ln(c)}{c-1} \cdot \lim\limits_{c\to 1} \;\frac{c^4-6c^2+1}{4c^2+4c}\right) =\exp\left(1\cdot\frac{-4}{8} \right)= \frac{1}{\sqrt{e}} $$ because $\lim\limits_{c\to 1} \;\frac{\ln(c)}{c-1}$ is the derivative of $\ln(x)$ evaluated at $x=1$, which is $1$.

Answer 6

We have that

$$\cot(x)^{\cot(4*x)}=\left[\left(1+(\cot(x)-1)\right)^{\frac1{\cot(x)-1}}\right]^{\cot (4x)(\cot(x)-1)} \to$$

indeed

$$\left(1+(\cot(x)-1)\right)^{\frac1{\cot(x)-1}}\to e^{-\frac12}$$

and by trigonometric identities

$$\cot (4x)(\cot(x)-1)=\frac{\cos^2 (2x)-\sin^2(2x)}{2\cos(2x)\sin (2x)}\left(\frac{1+\cos (2x)}{\sin (2x)}-1\right) \\=\frac{1-2\sin^2(2x)}{2\sin^2(2x)}\left(\frac{1+\cos (2x)-\sin(2x)}{\cos (2x)}\right)\to -\frac12$$

since by $y=\frac{\pi}2-2x \to 0$

$$\frac{1+\cos (2x)-\sin(2x)}{\cos (2x)}=\frac{1+\sin y-\cos y}{\sin y}=1+\frac{1-\cos y}{y^2}\frac{y}{\sin y}y\to 1+0=1$$