Question
Graph all asymptotes of the rational function.
Ask by Norton Mitchell. in the United States
Nov 14,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The function has a vertical asymptote at
and no horizontal asymptotes.
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To find the asymptotes of the function
, we start with vertical asymptotes. Vertical asymptotes occur where the denominator is zero, but the numerator is not. Setting the denominator equal to zero, we have:
So, there is a vertical asymptote at
.
Next, we find the horizontal asymptote. For rational functions, if the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is
. However, if the degrees are equal, as in our case where the degree of the numerator (2) is greater than that of the denominator (1), there is no horizontal asymptote. Instead, we check for an oblique/slant asymptote by performing polynomial long division.
Dividing
by
, we get:
- Divide the leading term:
- Multiply
by the entire divisor: - Subtract:
- Divide the leading term again:
- Multiply:
- Subtract:
Thus, we find that:
As
or
,
approaches
, indicating that the slant asymptote is
.
In summary:
- Vertical asymptote:
- Oblique/slant asymptote: