# What are the asymptotes of #g(x) = (x-1)/(x-x^3)#?

The vertical asymptotes are -1, 0 and 1, The horizontal asymptote is y=0.

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To find the asymptotes of ( g(x) = \frac{x - 1}{x - x^3} ), we first need to identify the vertical asymptotes, horizontal asymptotes, and oblique asymptotes, if any.

Vertical asymptotes occur where the denominator of a rational function becomes zero, excluding any values that would make the numerator zero as well. In this case, the denominator ( x - x^3 ) becomes zero when ( x = 0, 1, -1 ).

Horizontal asymptotes occur when the degree of the numerator is less than the degree of the denominator. In this case, since the degree of the numerator and the denominator are equal (both are 1), there are no horizontal asymptotes.

To check for any oblique asymptotes, divide the numerator by the denominator using polynomial long division. Upon performing the division, you'll find that there is no oblique asymptote for this function.

So, the vertical asymptotes of ( g(x) = \frac{x - 1}{x - x^3} ) are ( x = 0, 1, -1 ). There are no horizontal or oblique asymptotes.

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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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