Find the x-values (if any) at which f is not continuous. which of the discontinuities are removable? (use k as an arbitrary integer. if an answer does not exist, enter dne.) f (x) = csc 5x

Here we are to find the values of x in which the function f (x) = csc 5x is discontinuous. What this usually means is to find when the value approaches infinity.

We know from trigonometric identities that csc is the equivalent to 1 / sin, therefore the function in terms of sin is:

f (x) = 1 / sin 5x

We can see that the function becomes discontinuous when sin 5x = 0, that is a value divided by 0 is discontinuous or approaches infinity.

sin 5x is equal to zero when:

5x = 0 or π

So given k as an arbitrary integer

5x = k π

So k can be: k = 0, 1, 2

x = k π / 5