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cot(2*x)*tan(7*x)
Limit of the function/ cot(2*x)*tan(7*x)

Limit of the function cot(2*x)*tan(7*x)

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You have entered [src] 
 lim (cot(2*x)*tan(7*x))
x->0+
lim⁡x→0+(tan⁡(7x)cot⁡(2x))\lim_{x \to 0^+}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) 
Limit(cot(2*x)*tan(7*x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+tan⁡(7x)=0\lim_{x \to 0^+} \tan{\left(7 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+(tan⁡(7x)cot⁡(2x))\lim_{x \to 0^+}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right)
=
lim⁡x→0+(ddxtan⁡(7x)ddx1cot⁡(2x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(7 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(2 x \right)}}}\right)
=
lim⁡x→0+(7tan⁡2(7x)cot⁡2(2x)+7cot⁡2(2x)2cot⁡2(2x)+2)\lim_{x \to 0^+}\left(\frac{7 \tan^{2}{\left(7 x \right)} \cot^{2}{\left(2 x \right)} + 7 \cot^{2}{\left(2 x \right)}}{2 \cot^{2}{\left(2 x \right)} + 2}\right)
=
lim⁡x→0+(7tan⁡2(7x)cot⁡2(2x)+7cot⁡2(2x)2cot⁡2(2x)+2)\lim_{x \to 0^+}\left(\frac{7 \tan^{2}{\left(7 x \right)} \cot^{2}{\left(2 x \right)} + 7 \cot^{2}{\left(2 x \right)}}{2 \cot^{2}{\left(2 x \right)} + 2}\right)
=
72\frac{7}{2}
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
02468-8-6-4-2-1010-100100
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Rapid solution [src] 
7/2
72\frac{7}{2}
Expand and simplify
One‐sided limits [src] 
 lim (cot(2*x)*tan(7*x))
x->0+
lim⁡x→0+(tan⁡(7x)cot⁡(2x))\lim_{x \to 0^+}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) 
7/2
72\frac{7}{2} 
= 3.5
 lim (cot(2*x)*tan(7*x))
x->0-
lim⁡x→0−(tan⁡(7x)cot⁡(2x))\lim_{x \to 0^-}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) 
7/2
72\frac{7}{2} 
= 3.5
= 3.5
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(tan⁡(7x)cot⁡(2x))=72\lim_{x \to 0^-}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) = \frac{7}{2}
More at x→0 from the left
lim⁡x→0+(tan⁡(7x)cot⁡(2x))=72\lim_{x \to 0^+}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) = \frac{7}{2}
lim⁡x→∞(tan⁡(7x)cot⁡(2x))\lim_{x \to \infty}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right)
More at x→oo
lim⁡x→1−(tan⁡(7x)cot⁡(2x))=tan⁡(7)tan⁡(2)\lim_{x \to 1^-}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) = \frac{\tan{\left(7 \right)}}{\tan{\left(2 \right)}}
More at x→1 from the left
lim⁡x→1+(tan⁡(7x)cot⁡(2x))=tan⁡(7)tan⁡(2)\lim_{x \to 1^+}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right) = \frac{\tan{\left(7 \right)}}{\tan{\left(2 \right)}}
More at x→1 from the right
lim⁡x→−∞(tan⁡(7x)cot⁡(2x))\lim_{x \to -\infty}\left(\tan{\left(7 x \right)} \cot{\left(2 x \right)}\right)
More at x→-oo
Numerical answer [src] 
3.5
3.5
The graph
Limit of the function cot(2*x)*tan(7*x)
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