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(sin(x)+tan(x))/(2*x)
Limit of the function/ (sin(x)+tan(x))/(2*x)

Limit of the function (sin(x)+tan(x))/(2*x)

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     /sin(x) + tan(x)\
 lim |---------------|
x->0+\      2*x      /
lim⁡x→0+(sin⁡(x)+tan⁡(x)2x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) 
Limit((sin(x) + tan(x))/((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⁡(x)+tan⁡(x))=0\lim_{x \to 0^+}\left(\sin{\left(x \right)} + \tan{\left(x \right)}\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+(sin⁡(x)+tan⁡(x)2x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡(x)+tan⁡(x)2x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right)
=
lim⁡x→0+(ddx(sin⁡(x)+tan⁡(x))ddx2x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}{\frac{d}{d x} 2 x}\right)
=
lim⁡x→0+(cos⁡(x)2+tan⁡2(x)2+12)\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{2} + \frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)
=
lim⁡x→0+(cos⁡(x)2+tan⁡2(x)2+12)\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{2} + \frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)
=
11
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
02468-8-6-4-2-1010-1010
Plot the graph
One‐sided limits [src] 
     /sin(x) + tan(x)\
 lim |---------------|
x->0+\      2*x      /
lim⁡x→0+(sin⁡(x)+tan⁡(x)2x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) 
1
11 
= 1.0
     /sin(x) + tan(x)\
 lim |---------------|
x->0-\      2*x      /
lim⁡x→0−(sin⁡(x)+tan⁡(x)2x)\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) 
1
11 
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(x)+tan⁡(x)2x)=1\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) = 1
More at x→0 from the left
lim⁡x→0+(sin⁡(x)+tan⁡(x)2x)=1\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) = 1
lim⁡x→∞(sin⁡(x)+tan⁡(x)2x)\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right)
More at x→oo
lim⁡x→1−(sin⁡(x)+tan⁡(x)2x)=sin⁡(1)2+tan⁡(1)2\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) = \frac{\sin{\left(1 \right)}}{2} + \frac{\tan{\left(1 \right)}}{2}
More at x→1 from the left
lim⁡x→1+(sin⁡(x)+tan⁡(x)2x)=sin⁡(1)2+tan⁡(1)2\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right) = \frac{\sin{\left(1 \right)}}{2} + \frac{\tan{\left(1 \right)}}{2}
More at x→1 from the right
lim⁡x→−∞(sin⁡(x)+tan⁡(x)2x)\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x}\right)
More at x→-oo
Rapid solution [src] 
1
11
Expand and simplify
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function (sin(x)+tan(x))/(2*x)
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