Please explain all the steps please Consider the function fx

Please explain all the steps please.

Consider the function f(x)=(1-cos2x)/(x³-4x)

Find the domain

Is it an odd or even function?

Evaluate lim_x-->0f(x)

Classify all discontinuities of the function.

List all vertical asymptotes is any exist.

Is the function continuous at the point x=?

Solution

Domain :
x^3- 4x = 0
x(x^2 - 4) = 0
x = -2 , 0 and 2

So, dmain : {x is anything except -2 , 0 and 2}

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The numerator is even
denominator is odd

So, even/odd = odd

So, odd function

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We can use L Hospital\'s rule here...

Deriving :
f\'(x) = 2sin(2x) / (3x^2 -4)

Now, we apply limit x = 0
2sin(2*0) / (3*0 - 4)
2sin(0) / -4
0 is the limit

0

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Discontinuities :
x = -2 --> inf discontinuity
x = 0 --> same
x = 2 --> Same

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VA :
x = -2 , 0 and 2

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Yes, continuoius when x = pi
because when we sub in pi, denominator AINT 0