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(1/sin(x))^tan(x)
Limit of the function/ (1/sin(x))^tan(x)

Limit of the function (1/sin(x))^tan(x)

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             tan(x)
     /  1   \      
 lim |------|      
x->0+\sin(x)/
lim⁡x→0+(1sin⁡(x))tan⁡(x)\lim_{x \to 0^+} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} 
Limit((1/sin(x))^tan(x), x, 0)
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
02468-8-6-4-2-10100.02.0
Plot the graph
Rapid solution [src] 
1
11
Expand and simplify
One‐sided limits [src] 
             tan(x)
     /  1   \      
 lim |------|      
x->0+\sin(x)/
lim⁡x→0+(1sin⁡(x))tan⁡(x)\lim_{x \to 0^+} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} 
1
11 
= 1.00163490275634
             tan(x)
     /  1   \      
 lim |------|      
x->0-\sin(x)/
lim⁡x→0−(1sin⁡(x))tan⁡(x)\lim_{x \to 0^-} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} 
1
11 
= (0.998071492041113 - 0.000766527748721653j)
= (0.998071492041113 - 0.000766527748721653j)
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(1sin⁡(x))tan⁡(x)=1\lim_{x \to 0^-} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} = 1
More at x→0 from the left
lim⁡x→0+(1sin⁡(x))tan⁡(x)=1\lim_{x \to 0^+} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} = 1
lim⁡x→∞(1sin⁡(x))tan⁡(x)\lim_{x \to \infty} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}}
More at x→oo
lim⁡x→1−(1sin⁡(x))tan⁡(x)=sin⁡−tan⁡(1)(1)\lim_{x \to 1^-} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} = \sin^{- \tan{\left(1 \right)}}{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(1sin⁡(x))tan⁡(x)=sin⁡−tan⁡(1)(1)\lim_{x \to 1^+} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}} = \sin^{- \tan{\left(1 \right)}}{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(1sin⁡(x))tan⁡(x)\lim_{x \to -\infty} \left(\frac{1}{\sin{\left(x \right)}}\right)^{\tan{\left(x \right)}}
More at x→-oo
Numerical answer [src] 
1.00163490275634
1.00163490275634
The graph
Limit of the function (1/sin(x))^tan(x)
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