How to determine slant asymptote?
Understand the Problem
The question is asking for the methods or steps to find slant asymptotes of a function, particularly rational functions. Slant asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. The high-level approach includes polynomial long division to find the equation of the slant asymptote.
Answer
The slant asymptote can be found using polynomial long division resulting in $y = mx + b$.
Answer for screen readers
The method to find the slant asymptote of a function involves polynomial long division, yielding an equation of the form $y = mx + b$, where $m$ and $b$ are derived from the quotient of the division.
Steps to Solve
- Identify the degrees of the numerator and denominator
Check the degrees (the highest power of x) of both the numerator and the denominator. The function must be a rational function where the numerator's degree is one more than the denominator's degree for a slant asymptote to exist.
- Perform Polynomial Long Division
If the degree condition is satisfied, divide the numerator by the denominator using polynomial long division. This will help separate the polynomial part from the remainder.
- Extract the Leading Term
Take the result of the long division and focus on the polynomial part (the quotient). The leading term of this polynomial is the equation of the slant asymptote.
- Write the Equation of the Slant Asymptote
The equation of the slant asymptote can be written as: $$ y = mx + b $$ Where $m$ is the slope and $b$ is the y-intercept from the polynomial obtained in the long division.
- Verify with the Original Function
Check the behavior of the original function as $x$ approaches infinity. The function should approach the line given by the slant asymptote if it exists.
The method to find the slant asymptote of a function involves polynomial long division, yielding an equation of the form $y = mx + b$, where $m$ and $b$ are derived from the quotient of the division.
More Information
Slant asymptotes indicate the behavior of rational functions at infinity. They occur when the numerator is one degree higher than the denominator, which is a unique feature studied in calculus and algebra.
Tips
- Forgetting to check the degree condition before performing long division.
- Confusing slant asymptotes with vertical or horizontal asymptotes, which occur under different conditions.
- Incorrectly interpreting the remainder from the long division process.
