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

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

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

i.e. limit for the numerator is
lim⁡x→0+sin⁡(2x)=0\lim_{x \to 0^+} \sin{\left(2 x \right)} = 0
and limit for the denominator is
lim⁡x→0+1cot⁡(7x)=0\lim_{x \to 0^+} \frac{1}{\cot{\left(7 x \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡(2x)cot⁡(7x))\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right)
=
lim⁡x→0+(ddxsin⁡(2x)ddx1cot⁡(7x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(7 x \right)}}}\right)
=
lim⁡x→0+(2cos⁡(2x)cot⁡2(7x)7cot⁡2(7x)+7)\lim_{x \to 0^+}\left(\frac{2 \cos{\left(2 x \right)} \cot^{2}{\left(7 x \right)}}{7 \cot^{2}{\left(7 x \right)} + 7}\right)
=
lim⁡x→0+(2cot⁡2(7x)7cot⁡2(7x)+7)\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(7 x \right)}}{7 \cot^{2}{\left(7 x \right)} + 7}\right)
=
lim⁡x→0+(2cot⁡2(7x)7cot⁡2(7x)+7)\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(7 x \right)}}{7 \cot^{2}{\left(7 x \right)} + 7}\right)
=
27\frac{2}{7}
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
02468-8-6-4-2-1010-5050
Plot the graph
One‐sided limits [src] 
 lim (cot(7*x)*sin(2*x))
x->0+
lim⁡x→0+(sin⁡(2x)cot⁡(7x))\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right) 
2/7
27\frac{2}{7} 
= 0.285714285714286
 lim (cot(7*x)*sin(2*x))
x->0-
lim⁡x→0−(sin⁡(2x)cot⁡(7x))\lim_{x \to 0^-}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right) 
2/7
27\frac{2}{7} 
= 0.285714285714286
= 0.285714285714286
Rapid solution [src] 
2/7
27\frac{2}{7}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(2x)cot⁡(7x))=27\lim_{x \to 0^-}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right) = \frac{2}{7}
More at x→0 from the left
lim⁡x→0+(sin⁡(2x)cot⁡(7x))=27\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right) = \frac{2}{7}
lim⁡x→∞(sin⁡(2x)cot⁡(7x))\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right)
More at x→oo
lim⁡x→1−(sin⁡(2x)cot⁡(7x))=sin⁡(2)tan⁡(7)\lim_{x \to 1^-}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right) = \frac{\sin{\left(2 \right)}}{\tan{\left(7 \right)}}
More at x→1 from the left
lim⁡x→1+(sin⁡(2x)cot⁡(7x))=sin⁡(2)tan⁡(7)\lim_{x \to 1^+}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right) = \frac{\sin{\left(2 \right)}}{\tan{\left(7 \right)}}
More at x→1 from the right
lim⁡x→−∞(sin⁡(2x)cot⁡(7x))\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cot{\left(7 x \right)}\right)
More at x→-oo
Numerical answer [src] 
0.285714285714286
0.285714285714286
The graph
Limit of the function cot(7*x)*sin(2*x)
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