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cot(5*x)*tan(3*x)
Limit of the function/ cot(5*x)*tan(3*x)

Limit of the function cot(5*x)*tan(3*x)

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You have entered [src] 
 lim (cot(5*x)*tan(3*x))
x->0+
lim⁡x→0+(tan⁡(3x)cot⁡(5x))\lim_{x \to 0^+}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) 
Limit(cot(5*x)*tan(3*x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+tan⁡(3x)=0\lim_{x \to 0^+} \tan{\left(3 x \right)} = 0
and limit for the denominator is
lim⁡x→0+1cot⁡(5x)=0\lim_{x \to 0^+} \frac{1}{\cot{\left(5 x \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(tan⁡(3x)cot⁡(5x))\lim_{x \to 0^+}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right)
=
lim⁡x→0+(ddxtan⁡(3x)ddx1cot⁡(5x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(3 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(5 x \right)}}}\right)
=
lim⁡x→0+(3tan⁡2(3x)cot⁡2(5x)+3cot⁡2(5x)5cot⁡2(5x)+5)\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 3 \cot^{2}{\left(5 x \right)}}{5 \cot^{2}{\left(5 x \right)} + 5}\right)
=
lim⁡x→0+(3tan⁡2(3x)cot⁡2(5x)+3cot⁡2(5x)5cot⁡2(5x)+5)\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 3 \cot^{2}{\left(5 x \right)}}{5 \cot^{2}{\left(5 x \right)} + 5}\right)
=
35\frac{3}{5}
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
02468-8-6-4-2-1010-500500
Plot the graph
One‐sided limits [src] 
 lim (cot(5*x)*tan(3*x))
x->0+
lim⁡x→0+(tan⁡(3x)cot⁡(5x))\lim_{x \to 0^+}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) 
3/5
35\frac{3}{5} 
= 0.6
 lim (cot(5*x)*tan(3*x))
x->0-
lim⁡x→0−(tan⁡(3x)cot⁡(5x))\lim_{x \to 0^-}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) 
3/5
35\frac{3}{5} 
= 0.6
= 0.6
Rapid solution [src] 
3/5
35\frac{3}{5}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(tan⁡(3x)cot⁡(5x))=35\lim_{x \to 0^-}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) = \frac{3}{5}
More at x→0 from the left
lim⁡x→0+(tan⁡(3x)cot⁡(5x))=35\lim_{x \to 0^+}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) = \frac{3}{5}
lim⁡x→∞(tan⁡(3x)cot⁡(5x))\lim_{x \to \infty}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right)
More at x→oo
lim⁡x→1−(tan⁡(3x)cot⁡(5x))=tan⁡(3)tan⁡(5)\lim_{x \to 1^-}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) = \frac{\tan{\left(3 \right)}}{\tan{\left(5 \right)}}
More at x→1 from the left
lim⁡x→1+(tan⁡(3x)cot⁡(5x))=tan⁡(3)tan⁡(5)\lim_{x \to 1^+}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right) = \frac{\tan{\left(3 \right)}}{\tan{\left(5 \right)}}
More at x→1 from the right
lim⁡x→−∞(tan⁡(3x)cot⁡(5x))\lim_{x \to -\infty}\left(\tan{\left(3 x \right)} \cot{\left(5 x \right)}\right)
More at x→-oo
Numerical answer [src] 
0.6
0.6
The graph
Limit of the function cot(5*x)*tan(3*x)
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