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

Question

Elias Ackerman · Accepted Answer

The vertical asymptote is #x=0#
The slant asymptote is #y=x#
No horizontal asymptote

As you cannot divide by #0#, #x!=0# So #x=0# is a vertical asymptote. The degree of the numerator is #># the degree of the denominator, so we expect a slant asymptote. Let's simplify the #h(x)# #h(x)=(x^2-4)/x=x-4/x# Therefore, #y=x# is a slant asymptote. #lim_(x->+-oo)h(x)=lim_(x->+-oo)x=+-oo# graph{(y-(x^2-4)/x)(y-x)=0 [-11.25, 11.25, -5.63, 5.62]}

Anna Allen · Answer

undefined To find the asymptotes of the function $ h(x) = \frac{x^2 - 4}{x} $, we need to consider both vertical and horizontal asymptotes.

1. **Vertical Asymptotes**: Vertical asymptotes occur where the function is undefined, which in this case is when the denominator is equal to zero (since division by zero is undefined).

Set the denominator equal to zero and solve for $ x $:
$$ x = 0 $$

So, there is a vertical asymptote at $ x = 0 $.

2. **Horizontal Asymptotes**: Horizontal asymptotes occur as $ x $ approaches positive or negative infinity. To find horizontal asymptotes, we examine the behavior of the function as $ x $ becomes very large in magnitude.

As $ x $ approaches positive or negative infinity, the term $ \frac{x^2}{x} $ dominates the function, as the $ x^2 $ term grows much faster than $ x $. Therefore, the function approaches $ y = x $ as $ x $ approaches positive or negative infinity.

Thus, the function has a horizontal asymptote at $ y = x $.

In summary, the function $ h(x) = \frac{x^2 - 4}{x} $ has a vertical asymptote at $ x = 0 $ and a horizontal asymptote at $ y = x $.

Xavier Alonso · Answer

undefined To find the asymptotes of $ h(x) = \frac{{x^2 - 4}}{x} $:

1. Vertical asymptote(s): Set the denominator equal to zero and solve for $ x $.
   $$ x = 0 $$
   So, there is a vertical asymptote at $ x = 0 $.

2. Horizontal asymptote: Compare the degrees of the numerator and denominator.
   Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote.

3. Oblique (slant) asymptote: If the degree of the numerator is exactly one more than the degree of the denominator, there may be an oblique asymptote. In this case, the degrees differ by one, so there is a slant asymptote.
   To find the equation of the oblique asymptote, perform polynomial long division or use synthetic division.
   $$ (x^2 - 4) \div x = x - 4 $$
   So, the equation of the slant asymptote is $ y = x - 4 $.