Pregunta
- Consider
.
a. What are the asymptotes?
b. What is the
?
c. What is the
?
d. What is the
?
a. What are the asymptotes?
b. What is the
c. What is the
d. What is the
Ask by Bird Davison. in the United States
Jan 26,2025
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Respuesta verificada por el tutor
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a. Vertical asymptote at
and horizontal asymptote at
.
b. As
approaches
from the right,
approaches
.
c. As
approaches
from the left,
approaches
.
d. As
approaches infinity,
approaches
.
Solución

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The Deep Dive
To dive into the fascinating world of the function
, let’s look at its asymptotes! The vertical asymptote occurs where the denominator is zero, which is at
. For the horizontal asymptote, we’re interested in the behavior as
approaches infinity. As
grows very large, the function simplifies to
. So, the asymptotes are:
- Vertical asymptote at
- Horizontal asymptote at
Now, let’s tackle those limits!
For
, as we approach -3 from the right, the denominator becomes a very small positive number, while the numerator approaches
. Thus, the limit tends to
. Conversely, for
, approaching -3 from the left gives us a very small negative number in the denominator. This leads the limit to
.
Lastly, for
, both the numerator and denominator are dominated by the respective
and
terms, simplifying our limit to
as mentioned before. Overall, the limits reveal some fun and wild behavior near our asymptote!

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