# What is the range of the function #y = csc x#?

Let us look at some details.

We know

by taking the reciprocal,

(Note that we need to switch the direction of inequality.)

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The range of the function ( y = \csc(x) ) is the set of all possible values that the function can output. Since the cosecant function is the reciprocal of the sine function, its range will be the set of all real numbers except for values where the sine function is equal to zero, because division by zero is undefined.

The range of ( \sin(x) ) is from -1 to 1, inclusive. So, the values of ( \csc(x) ) will range from negative infinity to negative one, unioned with positive one to positive infinity, excluding where ( \sin(x) ) equals zero.

Therefore, the range of ( y = \csc(x) ) is ( (-\infty, -1] \cup [1, \infty) ), excluding the points where ( \sin(x) = 0 ).

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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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