# How do you find the asymptotes for #h(x)=(x^2-4) / x#?

vertical asymptote x = 0

oblique asymptote y=x

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation , set the denominator to zero.

hence : x = 0 is the asymptote

Horizontal asymptotes occur when the degree of the numerator is ≤ to the degree of the denominator. This is not so here , hence there are no horizontal asymptotes.

Oblique asymptotes occur when the degree of the numerator > than the degree of the denominator. This is the case here.

divide numerator by x

Here is the graph of the function. graph{(x^2-4)/x [-10, 10, -5, 5]}

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To find the asymptotes of the function ( h(x) = \frac{x^2 - 4}{x} ), we analyze the behavior of the function as ( x ) approaches positive and negative infinity. We look for vertical asymptotes where the function approaches positive or negative infinity, and horizontal asymptotes where the function approaches a constant value.

Vertical asymptotes occur where the denominator becomes zero and the function approaches positive or negative infinity. Setting the denominator ( x ) equal to zero, we find that there is a vertical asymptote at ( x = 0 ).

To find horizontal asymptotes, we examine the behavior of the function as ( x ) approaches positive and negative infinity. As ( x ) approaches positive or negative infinity, the term ( \frac{x^2}{x} ) dominates the expression, resulting in the function approaching ( x ) as ( x ) goes to infinity. Therefore, there is a horizontal asymptote at ( y = x ).

In summary:

- Vertical asymptote: ( x = 0 )
- Horizontal asymptote: ( y = x )

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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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