How do you identify all asymptotes or holes for #f(x)=1/(2x2)#?
The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is nonzero for this value then it is a vertical asymptote.
Horizontal asymptotes occur as
divide terms on numerator/denominator by x
Holes occur when there are duplicate factors on the numerator/denominator. This is not the case here hence there are no holes. graph{1/(2x2) [10, 10, 5, 5]}
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To identify all asymptotes or holes for ( f(x) = \frac{1}{2x  2} ), we first need to determine the behavior of the function as ( x ) approaches certain values.

Vertical Asymptote:
 Set the denominator equal to zero and solve for ( x ). This is because vertical asymptotes occur where the function approaches positive or negative infinity as ( x ) approaches a certain value.
 For ( f(x) = \frac{1}{2x  2} ), the denominator ( 2x  2 ) equals zero when ( x = 1 ).
 Thus, the vertical asymptote is at ( x = 1 ).

Horizontal Asymptote:
 To find the horizontal asymptote, examine the behavior of the function as ( x ) approaches positive or negative infinity.
 For rational functions like this one, the degree of the numerator and the degree of the denominator determine the horizontal asymptote.
 Since the degree of the numerator is 0 and the degree of the denominator is 1, the horizontal asymptote is at ( y = 0 ) (the ( x )axis).

Hole:
 To identify a hole, check if there are any common factors in the numerator and the denominator that can be canceled out.
 In this case, there are no common factors that can be canceled out, so there is no hole in the graph.
In summary:
 Vertical asymptote: ( x = 1 )
 Horizontal asymptote: ( y = 0 )
 No hole in the graph.
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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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