Chapter 4 | Applications of Derivatives
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x +5) (8+4 x ) .
4.28 Sketch a graph of f ( x ) = (3
Example 4.30 Sketching a Rational Function with an Oblique Asymptote
Sketch the graph of f ( x ) = x 2 ( x −1)
Solution Step 1. The domain of f is the set of all real numbers x except x =1. Step 2. Find the intercepts. We can see that when x =0, f ( x ) =0, so (0, 0) is the only intercept. Step 3. Evaluate the limits at infinity. Since the degree of the numerator is one more than the degree of the denominator, f must have an oblique asymptote. To find the oblique asymptote, use long division of polynomials to write f ( x ) = x 2 x −1 = x +1+ 1 x −1 . Since 1/( x −1)→0 as x →±∞, f ( x ) approaches the line y = x +1 as x →±∞. The line y = x +1 is an oblique asymptote for f . Step 4. To check for vertical asymptotes, look at where the denominator is zero. Here the denominator is zero at x =1. Looking at both one-sided limits as x →1, we find lim x →1 + x 2 x −1 =∞and lim x →1 − x 2 x −1 =−∞. Therefore, x =1 is a vertical asymptote, and we have determined the behavior of f as x approaches 1 from the right and the left.
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