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

Answer 1

The vertical asymptotes are #x=-1# and #x=3#
No slant asymptote
The horizontal asymptote is #y=0#
No holes

Let's factorise the denominator

#x^2-2x-3=(x+1)(x-3)#
The domain of #f(x)# is #D_(f(x))=RR-{-1,3} #
As you cannot divide by #0#, #x!=-1# and #x!=3#
The vertical asymptotes are #x=-1# and #x=3#
The degree of the numerator #<# than the degree of the denominator, thereis no slant asymptote
We calculate the limits of #f(x)#, we take only the terms of highest degree.
#lim_(x->-oo)f(x)=lim_(x->-oo)(2x)/x^2=lim_(x->-oo)2/x=0^(-)#
#lim_(x->+oo)f(x)=lim_(x->+oo)(2x)/x^2=lim_(x->+oo)2/x=0^(+)#
The horizontal asymptote is #y=0#

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

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Answer 2

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

  1. Factor the denominator: ( x^2 - 2x - 3 = (x - 3)(x + 1) ).
  2. Set the denominator equal to zero and solve for ( x ): ( x - 3 = 0 ) and ( x + 1 = 0 ). Therefore, ( x = 3 ) and ( x = -1 ).
  3. 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.
  4. 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.
  5. Therefore, the vertical asymptotes for ( f(x) ) are ( x = 3 ) and ( x = -1 ). There are no holes.
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Answer from HIX Tutor

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.

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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