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(-log(sin(x))+sin(x))/x
Limit of the function/ (-log(sin(x))+sin(x))/x

Limit of the function (-log(sin(x))+sin(x))/x

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