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4*x/sin(2*x)
Limit of the function/ 4*x/sin(2*x)

Limit of the function 4*x/sin(2*x)

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     /  4*x   \
 lim |--------|
x->oo\sin(2*x)/
$$\lim_{x \to \infty}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right)$$ 
Limit((4*x)/sin(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
Plot the graph
Rapid solution [src] 
     /  4*x   \
 lim |--------|
x->oo\sin(2*x)/
$$\lim_{x \to \infty}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right)$$
Expand and simplify
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right)$$
$$\lim_{x \to 0^-}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right) = 2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right) = 2$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right) = \frac{4}{\sin{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right) = \frac{4}{\sin{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4 x}{\sin{\left(2 x \right)}}\right)$$
More at x→-oo
The graph
Limit of the function 4*x/sin(2*x)
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