When is there no vertical asymptote?
Understand the Problem
The question is asking about the conditions under which a function will not have a vertical asymptote. Vertical asymptotes typically occur in rational functions where the denominator approaches zero. So, a function will not have a vertical asymptote when the denominator does not equal zero or is canceled out with a corresponding factor in the numerator.
Answer
When the zeros in the denominator cancel out with zeros in the numerator or when the denominator has no real zeros.
A rational function has no vertical asymptote when the zeros in the denominator are also zeros in the numerator and cancel out, or when setting the denominator to zero does not produce any real solutions.
Answer for screen readers
A rational function has no vertical asymptote when the zeros in the denominator are also zeros in the numerator and cancel out, or when setting the denominator to zero does not produce any real solutions.
More Information
A common situation where there are no vertical asymptotes is when a rational function has the same factor in its numerator and denominator, meaning they cancel out. Another situation is when the function's denominator does not result in real zeroes, such as when the denominator is always positive for real numbers.
Tips
A common mistake is to not fully simplify the rational function before determining vertical asymptotes. Always factor and cancel matching terms in the numerator and denominator first.
Sources
- Vertical Asymptotes Definition | Study.com - study.com
- Explain why (1)/(x^2+4) has no vertical asymptotes | Wyzant - wyzant.com
- How to find out whether a function has vertical asymptotes | Math Stack Exchange - math.stackexchange.com
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