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cot(2*x)/cot(8*x)
Limit of the function/ cot(2*x)/cot(8*x)

Limit of the function cot(2*x)/cot(8*x)

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     /cot(2*x)\
 lim |--------|
x->0+\cot(8*x)/
lim⁡x→0+(cot⁡(2x)cot⁡(8x))\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) 
Limit(cot(2*x)/cot(8*x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+1cot⁡(8x)=0\lim_{x \to 0^+} \frac{1}{\cot{\left(8 x \right)}} = 0
and limit for the denominator is
lim⁡x→0+1cot⁡(2x)=0\lim_{x \to 0^+} \frac{1}{\cot{\left(2 x \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(cot⁡(2x)cot⁡(8x))\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)
=
lim⁡x→0+(ddx1cot⁡(8x)ddx1cot⁡(2x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(8 x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(2 x \right)}}}\right)
=
lim⁡x→0+(8cot⁡2(2x)cot⁡2(8x)+8cot⁡2(2x)(2cot⁡2(2x)+2)cot⁡2(8x))\lim_{x \to 0^+}\left(\frac{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}}{\left(2 \cot^{2}{\left(2 x \right)} + 2\right) \cot^{2}{\left(8 x \right)}}\right)
=
lim⁡x→0+(ddx12cot⁡2(2x)+2ddxcot⁡2(8x)8cot⁡2(2x)cot⁡2(8x)+8cot⁡2(2x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(2 x \right)} + 2}}{\frac{d}{d x} \frac{\cot^{2}{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}}}\right)
=
lim⁡x→0+(−2(−4cot⁡2(2x)−4)cot⁡(2x)((−16cot⁡2(8x)−16)cot⁡(8x)8cot⁡2(2x)cot⁡2(8x)+8cot⁡2(2x)+(−8(−4cot⁡2(2x)−4)cot⁡(2x)cot⁡2(8x)−8(−4cot⁡2(2x)−4)cot⁡(2x)−8(−16cot⁡2(8x)−16)cot⁡2(2x)cot⁡(8x))cot⁡2(8x)(8cot⁡2(2x)cot⁡2(8x)+8cot⁡2(2x))2)(2cot⁡2(2x)+2)2)\lim_{x \to 0^+}\left(- \frac{2 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)}}{\left(\frac{\left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}} + \frac{\left(- 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} \cot^{2}{\left(8 x \right)} - 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} - 8 \left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot^{2}{\left(2 x \right)} \cot{\left(8 x \right)}\right) \cot^{2}{\left(8 x \right)}}{\left(8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}\right)^{2}}\right) \left(2 \cot^{2}{\left(2 x \right)} + 2\right)^{2}}\right)
=
lim⁡x→0+(−2(−4cot⁡2(2x)−4)cot⁡(2x)((−16cot⁡2(8x)−16)cot⁡(8x)8cot⁡2(2x)cot⁡2(8x)+8cot⁡2(2x)+(−8(−4cot⁡2(2x)−4)cot⁡(2x)cot⁡2(8x)−8(−4cot⁡2(2x)−4)cot⁡(2x)−8(−16cot⁡2(8x)−16)cot⁡2(2x)cot⁡(8x))cot⁡2(8x)(8cot⁡2(2x)cot⁡2(8x)+8cot⁡2(2x))2)(2cot⁡2(2x)+2)2)\lim_{x \to 0^+}\left(- \frac{2 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)}}{\left(\frac{\left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}} + \frac{\left(- 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} \cot^{2}{\left(8 x \right)} - 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} - 8 \left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot^{2}{\left(2 x \right)} \cot{\left(8 x \right)}\right) \cot^{2}{\left(8 x \right)}}{\left(8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}\right)^{2}}\right) \left(2 \cot^{2}{\left(2 x \right)} + 2\right)^{2}}\right)
=
44
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
02468-8-6-4-2-1010-100100
Plot the graph
Rapid solution [src] 
4
44
Expand and simplify
One‐sided limits [src] 
     /cot(2*x)\
 lim |--------|
x->0+\cot(8*x)/
lim⁡x→0+(cot⁡(2x)cot⁡(8x))\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) 
4
44 
= 4.0
     /cot(2*x)\
 lim |--------|
x->0-\cot(8*x)/
lim⁡x→0−(cot⁡(2x)cot⁡(8x))\lim_{x \to 0^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) 
4
44 
= 4.0
= 4.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(cot⁡(2x)cot⁡(8x))=4\lim_{x \to 0^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = 4
More at x→0 from the left
lim⁡x→0+(cot⁡(2x)cot⁡(8x))=4\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = 4
lim⁡x→∞(cot⁡(2x)cot⁡(8x))\lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)
More at x→oo
lim⁡x→1−(cot⁡(2x)cot⁡(8x))=tan⁡(8)tan⁡(2)\lim_{x \to 1^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\tan{\left(2 \right)}}
More at x→1 from the left
lim⁡x→1+(cot⁡(2x)cot⁡(8x))=tan⁡(8)tan⁡(2)\lim_{x \to 1^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\tan{\left(2 \right)}}
More at x→1 from the right
lim⁡x→−∞(cot⁡(2x)cot⁡(8x))\lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)
More at x→-oo
Numerical answer [src] 
4.0
4.0
The graph
Limit of the function cot(2*x)/cot(8*x)
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