Find the equation of the slant asymptote of the function [tex]f(x)=\frac{x^2+x+4}{x-1}[/tex]
A) [tex]y=x-2[/tex]
B) [tex]y=x+2[/tex]
C) [tex]y=x+6 /(x-1)[/tex]
Answer (1)
Perform polynomial long division of x 2 + x + 4 by x − 1 .
Identify the quotient from the long division, which is x + 2 .
The slant asymptote is given by the equation y = x + 2 .
The equation of the slant asymptote is y = x + 2 .
Explanation
Understanding the Problem We are asked to find the slant asymptote of the function f ( x ) = x − 1 x 2 + x + 4 . A slant asymptote occurs when the degree of the numerator is one greater than the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 1, so a slant asymptote exists.
Finding the Quotient To find the slant asymptote, we perform polynomial long division to divide the numerator by the denominator. This will give us an expression of the form f ( x ) = q ( x ) + x − 1 r ( x ) , where q ( x ) is the quotient and r ( x ) is the remainder. The slant asymptote is then given by the equation y = q ( x ) .
Polynomial Long Division Performing polynomial long division of x 2 + x + 4 by x − 1 , we get:
\multicolumn 2 r x \cline 2 − 5 x − 1 \multicolumn 2 r x 2 \cline 2 − 3 \multicolumn 2 r 0 \multicolumn 2 r \cline 3 − 4 \multicolumn 2 r + 2 x 2 − x 2 x 2 x 0 + x + 4 − 2 6 + 4
So, we have x 2 + x + 4 = ( x − 1 ) ( x + 2 ) + 6 . Therefore, f ( x ) = x + 2 + x − 1 6 .
Identifying the Slant Asymptote The quotient is x + 2 , and the remainder is 6. Thus, the slant asymptote is given by the equation y = x + 2 .
Final Answer Therefore, the equation of the slant asymptote is y = x + 2 .
Examples
Slant asymptotes are useful in various real-world applications, such as analyzing the behavior of rational functions that model population growth, chemical concentrations, or electrical circuits. For example, if a function represents the concentration of a chemical in a reactor over time, the slant asymptote can indicate the long-term trend of the concentration as time increases. Understanding slant asymptotes helps engineers and scientists predict and control the behavior of these systems.
