Math Problem Statement
Solution
We are given the function and asked to find the horizontal and vertical asymptotes using limits.
1. Horizontal Asymptotes:
To find the horizontal asymptote, we examine the behavior of as and .
As or , the highest degree terms (those involving ) dominate in both the numerator and the denominator. So, we can approximate:
Thus, the horizontal asymptote is:
2. Vertical Asymptotes:
Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. The denominator is .
Therefore, there are no vertical asymptotes.
3. Limits to Confirm Behavior:
- As , .
- As , .
These confirm the horizontal asymptote at .
There are no vertical asymptotes since the denominator never equals zero for real values of .
Would you like more details or have any questions?
Related Questions:
- How would the function change if there were a term like in the denominator?
- How do slant asymptotes differ from horizontal asymptotes?
- What if the degree of the numerator was higher than the denominator?
- How can the limits help us determine end behavior for functions with more complex exponents?
- Can there be horizontal asymptotes when the numerator and denominator have different degrees?
Tip:
Always compare the degrees of the numerator and denominator to quickly assess the horizontal asymptote in rational functions.
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Math Problem Analysis
Mathematical Concepts
Asymptotes
Limits
Rational Functions
Formulas
Horizontal asymptote: lim(x -> ∞) f(x)
Vertical asymptote: Set denominator = 0 and solve for x
Theorems
Limit Theorem for Rational Functions
Dominance of Highest Degree Terms
Suitable Grade Level
Grades 10-12
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