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cos(2*x)^3
Limit of the function/ cos(2*x)^3

Limit of the function cos(2*x)^3

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        3     
 lim cos (2*x)
x->0+
lim⁡x→0+cos⁡3(2x)\lim_{x \to 0^+} \cos^{3}{\left(2 x \right)} 
Limit(cos(2*x)^3, 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
02468-8-6-4-2-10102-2
Plot the graph
Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−cos⁡3(2x)=1\lim_{x \to 0^-} \cos^{3}{\left(2 x \right)} = 1
More at x→0 from the left
lim⁡x→0+cos⁡3(2x)=1\lim_{x \to 0^+} \cos^{3}{\left(2 x \right)} = 1
lim⁡x→∞cos⁡3(2x)=⟨−1,1⟩\lim_{x \to \infty} \cos^{3}{\left(2 x \right)} = \left\langle -1, 1\right\rangle
More at x→oo
lim⁡x→1−cos⁡3(2x)=cos⁡3(2)\lim_{x \to 1^-} \cos^{3}{\left(2 x \right)} = \cos^{3}{\left(2 \right)}
More at x→1 from the left
lim⁡x→1+cos⁡3(2x)=cos⁡3(2)\lim_{x \to 1^+} \cos^{3}{\left(2 x \right)} = \cos^{3}{\left(2 \right)}
More at x→1 from the right
lim⁡x→−∞cos⁡3(2x)=⟨−1,1⟩\lim_{x \to -\infty} \cos^{3}{\left(2 x \right)} = \left\langle -1, 1\right\rangle
More at x→-oo
One‐sided limits [src] 
        3     
 lim cos (2*x)
x->0+
lim⁡x→0+cos⁡3(2x)\lim_{x \to 0^+} \cos^{3}{\left(2 x \right)} 
1
11 
= 1.0
        3     
 lim cos (2*x)
x->0-
lim⁡x→0−cos⁡3(2x)\lim_{x \to 0^-} \cos^{3}{\left(2 x \right)} 
1
11 
= 1.0
= 1.0
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function cos(2*x)^3
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