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

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

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     /   3     \
     |cos (2*x)|
 lim |---------|
x->0+\    x    /
lim⁡x→0+(cos⁡3(2x)x)\lim_{x \to 0^+}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) 
Limit(cos(2*x)^3/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
02468-8-6-4-2-1010-200200
Plot the graph
Rapid solution [src] 
oo
∞\infty
Expand and simplify
One‐sided limits [src] 
     /   3     \
     |cos (2*x)|
 lim |---------|
x->0+\    x    /
lim⁡x→0+(cos⁡3(2x)x)\lim_{x \to 0^+}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) 
oo
∞\infty 
= 150.960268966734
     /   3     \
     |cos (2*x)|
 lim |---------|
x->0-\    x    /
lim⁡x→0−(cos⁡3(2x)x)\lim_{x \to 0^-}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) 
-oo
−∞-\infty 
= -150.960268966734
= -150.960268966734
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(cos⁡3(2x)x)=∞\lim_{x \to 0^-}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = \infty
More at x→0 from the left
lim⁡x→0+(cos⁡3(2x)x)=∞\lim_{x \to 0^+}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = \infty
lim⁡x→∞(cos⁡3(2x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = 0
More at x→oo
lim⁡x→1−(cos⁡3(2x)x)=cos⁡3(2)\lim_{x \to 1^-}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = \cos^{3}{\left(2 \right)}
More at x→1 from the left
lim⁡x→1+(cos⁡3(2x)x)=cos⁡3(2)\lim_{x \to 1^+}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = \cos^{3}{\left(2 \right)}
More at x→1 from the right
lim⁡x→−∞(cos⁡3(2x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = 0
More at x→-oo
Numerical answer [src] 
150.960268966734
150.960268966734
The graph
Limit of the function cos(2*x)^3/x
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