How do you graph #f(x)=(2x^32x^2)/(x^39x)# using holes, vertical and horizontal asymptotes, x and y intercepts?
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I personally wouldn't dare to end it here. Let's find how f(x) behaves in undefined points.
don't forget undefined point [0,0]
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To graph the function f(x) = (2x^3  2x^2)/(x^3  9x), we can analyze its holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

Holes: To find the holes, we need to factor the numerator and denominator and cancel out any common factors. In this case, we can factor out an x^2 from both terms in the numerator, and an x from both terms in the denominator. This simplifies the function to f(x) = (2x^2(x  1))/(x(x^2  9)). We can see that there is a hole at x = 0, as the x term cancels out.

Vertical Asymptotes: Vertical asymptotes occur when the denominator of a rational function equals zero. In this case, the denominator x(x^2  9) equals zero when x = 0 and x = ±3. Therefore, there are vertical asymptotes at x = 0 and x = ±3.

Horizontal Asymptotes: To determine the horizontal asymptotes, we compare the degrees of the numerator and denominator. In this case, the degree of the numerator (2x^2) is less than the degree of the denominator (x^3  9x). Therefore, there is a horizontal asymptote at y = 0 (the xaxis).

xintercepts: To find the xintercepts, we set f(x) = 0 and solve for x. In this case, we have 2x^2(x  1) = 0. This gives us x = 0 and x = 1 as the xintercepts.

yintercept: To find the yintercept, we set x = 0 in the function f(x). This gives us f(0) = 0, so the yintercept is at the origin (0,0).
By considering these aspects, we can plot the graph of f(x) = (2x^3  2x^2)/(x^3  9x) accordingly.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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