# How do you identify all asymptotes or holes for #f(x)=(2x-2)/(x^2-2x-3)#?

The vertical asymptotes are

No slant asymptote

The horizontal asymptote is

No holes

Let's factorise the denominator

graph{(2x-2)/(x^2-2x-3) [-8.89, 8.89, -4.444, 4.445]}

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To identify all asymptotes or holes for ( f(x) = \frac{2x - 2}{x^2 - 2x - 3} ), follow these steps:

- Factor the denominator: ( x^2 - 2x - 3 = (x - 3)(x + 1) ).
- Set the denominator equal to zero and solve for ( x ): ( x - 3 = 0 ) and ( x + 1 = 0 ). Therefore, ( x = 3 ) and ( x = -1 ).
- Determine if there are any vertical asymptotes by checking for values of ( x ) that make the denominator zero, excluding any values that also make the numerator zero. Here, ( x = 3 ) and ( x = -1 ) are the values that make the denominator zero.
- To find any holes, factor the numerator and denominator and cancel any common factors. If after cancellation, there is still a zero in both the numerator and denominator for some ( x ), then there is a hole. In this case, the numerator ( 2x - 2 ) does not have any common factors with the denominator after factoring, so there are no holes.
- Therefore, the vertical asymptotes for ( f(x) ) are ( x = 3 ) and ( x = -1 ). There are no holes.

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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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