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log(x)^sin(x)
Limit of the function/ log(x)^sin(x)

Limit of the function log(x)^sin(x)

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        sin(x)   
 lim log      (x)
x->0+
lim⁡x→0+log⁡(x)sin⁡(x)\lim_{x \to 0^+} \log{\left(x \right)}^{\sin{\left(x \right)}} 
Limit(log(x)^sin(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
02468-8-6-4-2-101004
Plot the graph
Rapid solution [src] 
1
11
Expand and simplify
One‐sided limits [src] 
        sin(x)   
 lim log      (x)
x->0+
lim⁡x→0+log⁡(x)sin⁡(x)\lim_{x \to 0^+} \log{\left(x \right)}^{\sin{\left(x \right)}} 
1
11 
= (1.00048305523721 + 0.000791898618241586j)
        sin(x)   
 lim log      (x)
x->0-
lim⁡x→0−log⁡(x)sin⁡(x)\lim_{x \to 0^-} \log{\left(x \right)}^{\sin{\left(x \right)}} 
1
11 
= (0.999471074028876 - 0.000663897284289529j)
= (0.999471074028876 - 0.000663897284289529j)
Other limits x→0, -oo, +oo, 1
lim⁡x→0−log⁡(x)sin⁡(x)=1\lim_{x \to 0^-} \log{\left(x \right)}^{\sin{\left(x \right)}} = 1
More at x→0 from the left
lim⁡x→0+log⁡(x)sin⁡(x)=1\lim_{x \to 0^+} \log{\left(x \right)}^{\sin{\left(x \right)}} = 1
lim⁡x→∞log⁡(x)sin⁡(x)=∞⟨−1,1⟩\lim_{x \to \infty} \log{\left(x \right)}^{\sin{\left(x \right)}} = \infty^{\left\langle -1, 1\right\rangle}
More at x→oo
lim⁡x→1−log⁡(x)sin⁡(x)=0\lim_{x \to 1^-} \log{\left(x \right)}^{\sin{\left(x \right)}} = 0
More at x→1 from the left
lim⁡x→1+log⁡(x)sin⁡(x)=0\lim_{x \to 1^+} \log{\left(x \right)}^{\sin{\left(x \right)}} = 0
More at x→1 from the right
lim⁡x→−∞log⁡(x)sin⁡(x)=∞⟨−1,1⟩\lim_{x \to -\infty} \log{\left(x \right)}^{\sin{\left(x \right)}} = \infty^{\left\langle -1, 1\right\rangle}
More at x→-oo
Numerical answer [src] 
(1.00048305523721 + 0.000791898618241586j)
(1.00048305523721 + 0.000791898618241586j)
The graph
Limit of the function log(x)^sin(x)
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