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x/(1-csc(x))
Limit of the function/ x/(1-csc(x))

Limit of the function x/(1-csc(x))

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The solution

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     /    x     \
 lim |----------|
x->0+\1 - csc(x)/
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \csc{\left(x \right)}}\right)$$ 
Limit(x/(1 - csc(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
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Rapid solution [src] 
0
$$0$$
Expand and simplify
One‐sided limits [src] 
     /    x     \
 lim |----------|
x->0+\1 - csc(x)/
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \csc{\left(x \right)}}\right)$$ 
0
$$0$$ 
= -1.63492060334732e-32
     /    x     \
 lim |----------|
x->0-\1 - csc(x)/
$$\lim_{x \to 0^-}\left(\frac{x}{1 - \csc{\left(x \right)}}\right)$$ 
0
$$0$$ 
= 2.17848766691616e-29
= 2.17848766691616e-29
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x}{1 - \csc{\left(x \right)}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \csc{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{1 - \csc{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{1 - \csc{\left(x \right)}}\right) = \frac{\sin{\left(1 \right)}}{-1 + \sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{1 - \csc{\left(x \right)}}\right) = \frac{\sin{\left(1 \right)}}{-1 + \sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{1 - \csc{\left(x \right)}}\right) = -\infty$$
More at x→-oo
Numerical answer [src] 
-1.63492060334732e-32
-1.63492060334732e-32
The graph
Limit of the function x/(1-csc(x))
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