How do you identify all asymptotes or holes for #f(x)=x/(4x^216x)#?
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.
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To identify all asymptotes or holes for the function ( f(x) = \frac{x}{4x^2  16x} ), we first need to analyze the behavior of the function as ( x ) approaches certain values.

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. To find them, we solve the equation ( 4x^2  16x = 0 ) for ( x ). This yields the values ( x = 0 ) and ( x = 4 ). Therefore, the vertical asymptotes are at ( x = 0 ) and ( x = 4 ).

Horizontal Asymptotes: Horizontal asymptotes occur when ( x ) approaches positive or negative infinity. To find them, we examine the behavior of the function as ( x ) approaches infinity. By observing the highest power terms in the numerator and denominator, we can see that as ( x ) approaches infinity, the term ( 4x^2 ) in the denominator dominates. Thus, the horizontal asymptote is at ( y = 0 ).

Holes: Holes occur when there is a factor common to both the numerator and the denominator of the function, resulting in a removable discontinuity. To find them, we factor both the numerator and the denominator and cancel any common factors. In this case, the numerator ( x ) and the denominator ( 4x(x + 4) ) share a common factor of ( x ). Canceling this common factor, we get ( f(x) = \frac{1}{4(x + 4)} ). This function has no real zeros in the numerator that would cause a hole. Hence, there are no holes in this function.
Therefore, the identified asymptotes for the function ( f(x) = \frac{x}{4x^2  16x} ) are:
 Vertical asymptotes at ( x = 0 ) and ( x = 4 ).
 Horizontal asymptote at ( y = 0 ).
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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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