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MTH 163 Asymptotes Tutorial

MTH 163 Asymptotes Tutorial - ASYMPTOTES TUTORIAL...

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ASYMPTOTES TUTORIAL Horizontal Vertical Slant and Holes
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Definition of an asymptote An asymptote is a straight line which acts as a boundary for the graph of a function. When a function has an asymptote (and not all functions have them) the function gets closer and closer to the asymptote as the input value to the function approaches either a specific value a or positive or negative infinity. The functions most likely to have asymptotes are rational functions
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Vertical Asymptotes Vertical asymptotes occur when the following condition is met: The denominator of the simplified rational function is equal to 0. Remember, the simplified rational function has cancelled any factors common to both the numerator and denominator.
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Finding Vertical Asymptotes Example 1 Given the function The first step is to cancel any factors common to both numerator and denominator. In this case there are none. The second step is to see where the denominator of the simplified function equals 0. ( 29 x x x f 2 2 5 2 + - = ( 29 1 0 1 0 1 2 0 2 2 - = = + = + = + x x x x
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Finding Vertical Asymptotes Example 1 Con’t. The vertical line x = -1 is the only vertical asymptote for the function. As the input value x to this function gets closer and closer to -1 the function itself looks and acts more and more like the vertical line x = -1.
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Graph of Example 1 The vertical dotted line at x = –1 is the vertical asymptote.
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Finding Vertical Asymptotes Example 2 If First simplify the function. Factor both numerator and denominator and cancel any common factors. ( 29 9 12 10 2 2 2 - + + = x x x x f ( 29 ( 29 ( 29 ( 29 3 4 2 3 3 4 2 3 9 12 10 2 3 2 - + = - + + + = - + + x x x x x x x x x
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Finding Vertical Asymptotes Example 2 Con’t.
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