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tan(6*x)/(2*x)
Limit of the function/ tan(6*x)/(2*x)

Limit of the function tan(6*x)/(2*x)

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     /tan(6*x)\
 lim |--------|
x->0+\  2*x   /
lim⁡x→0+(tan⁡(6x)2x)\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) 
Limit(tan(6*x)/((2*x)), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(tan⁡(6x)2x)\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right)
transform
lim⁡x→0+(tan⁡(6x)2x)=lim⁡x→0+(12xsin⁡(6x)cos⁡(6x))\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) = \lim_{x \to 0^+}\left(\frac{\frac{1}{2 x} \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}}\right)
=
lim⁡x→0+(12xsin⁡(6x))lim⁡x→0+1cos⁡(6x)=lim⁡x→0+(12xsin⁡(6x))\lim_{x \to 0^+}\left(\frac{1}{2 x} \sin{\left(6 x \right)}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(6 x \right)}} = \lim_{x \to 0^+}\left(\frac{1}{2 x} \sin{\left(6 x \right)}\right)
Do replacement
u=6xu = 6 x
then
lim⁡x→0+(sin⁡(6x)2x)=lim⁡u→0+(3sin⁡(u)u)\lim_{x \to 0^+}\left(\frac{\sin{\left(6 x \right)}}{2 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{u}\right)
=
3lim⁡u→0+(sin⁡(u)u)3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)
The limit
lim⁡u→0+(sin⁡(u)u)\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)
is first remarkable limit, is equal to 1.

The final answer:
lim⁡x→0+(tan⁡(6x)2x)=3\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) = 3
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+tan⁡(6x)=0\lim_{x \to 0^+} \tan{\left(6 x \right)} = 0
and limit for the denominator is
lim⁡x→0+(2x)=0\lim_{x \to 0^+}\left(2 x\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(tan⁡(6x)2x)\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(tan⁡(6x)2x)\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right)
=
lim⁡x→0+(ddxtan⁡(6x)ddx2x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(6 x \right)}}{\frac{d}{d x} 2 x}\right)
=
lim⁡x→0+(3tan⁡2(6x)+3)\lim_{x \to 0^+}\left(3 \tan^{2}{\left(6 x \right)} + 3\right)
=
lim⁡x→0+(3tan⁡2(6x)+3)\lim_{x \to 0^+}\left(3 \tan^{2}{\left(6 x \right)} + 3\right)
=
33
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
02468-8-6-4-2-1010-2020
Plot the graph
One‐sided limits [src] 
     /tan(6*x)\
 lim |--------|
x->0+\  2*x   /
lim⁡x→0+(tan⁡(6x)2x)\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) 
3
33 
= 3.0
     /tan(6*x)\
 lim |--------|
x->0-\  2*x   /
lim⁡x→0−(tan⁡(6x)2x)\lim_{x \to 0^-}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) 
3
33 
= 3.0
= 3.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(tan⁡(6x)2x)=3\lim_{x \to 0^-}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) = 3
More at x→0 from the left
lim⁡x→0+(tan⁡(6x)2x)=3\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) = 3
lim⁡x→∞(tan⁡(6x)2x)\lim_{x \to \infty}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right)
More at x→oo
lim⁡x→1−(tan⁡(6x)2x)=tan⁡(6)2\lim_{x \to 1^-}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) = \frac{\tan{\left(6 \right)}}{2}
More at x→1 from the left
lim⁡x→1+(tan⁡(6x)2x)=tan⁡(6)2\lim_{x \to 1^+}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right) = \frac{\tan{\left(6 \right)}}{2}
More at x→1 from the right
lim⁡x→−∞(tan⁡(6x)2x)\lim_{x \to -\infty}\left(\frac{\tan{\left(6 x \right)}}{2 x}\right)
More at x→-oo
Rapid solution [src] 
3
33
Expand and simplify
Numerical answer [src] 
3.0
3.0
The graph
Limit of the function tan(6*x)/(2*x)
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