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

Question

Sadie Alexander · Accepted Answer

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 # ((x^2)/x - 4/x) = x - 4/x # As# x tooo , 4/x to 0 " and " y to x # #rArr y = x " is the asymptote " # Here is the graph of the function. graph{(x^2-4)/x [-10, 10, -5, 5]}

Lincoln Anderson · Answer

undefined 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 $