248 rational functions 3x x2 4 solution we follow the

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Unformatted text preview: can be ‘plugged’ when embarking on a more advanced study of continuity. 5 236 Rational Functions Neither x = −1 nor x = 1 are in the domain of h, yet we see the behavior of the graph of y = h(x) is drastically different near these points. The reason for this lies in the second to last step when we x−1)(x−1) simplified the formula for h(x) in Example 4.1.1. We had h(x) = (2x+1)(x−1) . The reason x = −1 is ( not in the domain of h is because the factor (x + 1) appears in the denominator of h(x); similarly, x = 1 is not in the domain of h because of the factor (x − 1) in the denominator of h(x). The major difference between these two factors is that (x − 1) cancels with a factor in the numerator whereas (x + 1) does not. Loosely speaking, the trouble caused by (x − 1) in the denominator is canceled away while the factor (x + 1) remains to cause mischief. This is why the graph of y = h(x) has a vertical asymptote at x = −1 but only a hole at x = 1. These observations are generalized and summarized in the theorem below, whose proof is found in Calculus. Theorem 4.1. Location o...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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