What is the domain of #f(x)=cscx#?

Answer 1
The question gets much easier if we recall that, by definition, the function you're dealing with is #\frac{1}{\sin(x)}#.

Being a fraction, we have to make sure that the denominator is non-zero; and since there are no roots or logarithms, it is the only thing we have to study.

The domain of the function will thus be the following set: #\{ x \in \mathbb{R} | \sin(x)\ne0\}#
The sine function is defined as the projection of a point on the unit circle on the #y# axis, and thus #\sin(x)=0# if and only if the point belongs to the #x# axis.
The only two points which are on both the unit circle and on the #x# axis are #(1,0)# and #(-1,0)#, and they are given by a rotation of 0 and #\pi# radiants. Because of the periodicity of the sine function, 0 radiants is the same as #2k\pi# radians, and #pi# radians are the same as #(2k+1)\pi# radians, for each #k \in \mathbb{Z}#.
Finally, our answer is ready: we need to exclude from the domain all the points of the form #2k\pi#, for each #k \in \mathbb{Z}#. Using a proper notation, the domain is the set #D_f={x \in \mathbb{R} | x \ne 2k\pi, \forall k \in \mathbb{Z}#}
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Answer 2

The domain of ( f(x) = \csc(x) ) is ( x \neq k\pi ), where ( k ) is an integer, because the cosecant function is undefined at points where the sine function is zero. Therefore, the domain of ( f(x) = \csc(x) ) is ( x \neq k\pi ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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