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log(cos(2*x))
Limit of the function/ log(cos(2*x))

Limit of the function log(cos(2*x))

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You have entered [src] 
 lim log(cos(2*x))
x->oo
lim⁡x→∞log⁡(cos⁡(2x))\lim_{x \to \infty} \log{\left(\cos{\left(2 x \right)} \right)} 
Limit(log(cos(2*x)), x, oo, dir='-')
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
02468-8-6-4-2-10102.5-2.5
Plot the graph
Rapid solution [src] 
log(<-1, 1>)
log⁡(⟨−1,1⟩)\log{\left(\left\langle -1, 1\right\rangle \right)}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞log⁡(cos⁡(2x))=log⁡(⟨−1,1⟩)\lim_{x \to \infty} \log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→0−log⁡(cos⁡(2x))=0\lim_{x \to 0^-} \log{\left(\cos{\left(2 x \right)} \right)} = 0
More at x→0 from the left
lim⁡x→0+log⁡(cos⁡(2x))=0\lim_{x \to 0^+} \log{\left(\cos{\left(2 x \right)} \right)} = 0
More at x→0 from the right
lim⁡x→1−log⁡(cos⁡(2x))=log⁡(−cos⁡(2))+iπ\lim_{x \to 1^-} \log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(- \cos{\left(2 \right)} \right)} + i \pi
More at x→1 from the left
lim⁡x→1+log⁡(cos⁡(2x))=log⁡(−cos⁡(2))+iπ\lim_{x \to 1^+} \log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(- \cos{\left(2 \right)} \right)} + i \pi
More at x→1 from the right
lim⁡x→−∞log⁡(cos⁡(2x))=log⁡(⟨−1,1⟩)\lim_{x \to -\infty} \log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
More at x→-oo
The graph
Limit of the function log(cos(2*x))
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