Question

mptotic behavior of rational functions
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current objective
identify slant asymptotes of rational functions
question
find the equation of the slant asymptote for the function ( f(x)=\frac{2 x^{2}-5 x}{x - 2} ).
provide your answer below:
( y=square )

Explanation:

Step1: Divide the numerator by the denominator

We use polynomial long - division. Divide \(2x^{2}-5x\) by \(x - 2\).
First, divide the leading term of the numerator \(2x^{2}\) by the leading term of the denominator \(x\): \(\frac{2x^{2}}{x}=2x\).
Multiply \(x - 2\) by \(2x\): \(2x(x - 2)=2x^{2}-4x\).
Subtract this from the numerator: \((2x^{2}-5x)-(2x^{2}-4x)=2x^{2}-5x - 2x^{2}+4x=-x\).
Now, divide the new leading term \(-x\) by \(x\): \(\frac{-x}{x}=-1\).
Multiply \(x - 2\) by \(-1\): \(-1(x - 2)=-x + 2\).
Subtract this from \(-x\): \((-x)-(-x + 2)=-x+x - 2=-2\).
So, \(\frac{2x^{2}-5x}{x - 2}=2x-1-\frac{2}{x - 2}\).

Step2: Determine the slant asymptote

As \(x\to\pm\infty\), \(\frac{-2}{x - 2}\to0\).

Answer:

\(y = 2x-1\)