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"Defined items" are defined in terms of "undefined terms".
A slope is 0 if it is a straight horizontal line. A slope is undefined if it is a straight vertical line.
That is correct.
You have to include the expression in the question.
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You do not. If you do, the expression is undefined: it has no value.
The expression X/(X+8) is undefined at X=-8, because that would be division by zero.
A rational expression is not defined whenever the denominator of the expression equals zero. These will be the roots or zeros of the denominator.
A fraction such that the divisor (denominator) is zero is undefined. Such a division is expressed as x/0 where x is the dividend (numerator). In ordinary arithmetic, the expression has no meaning so division by zero is undefined.
The undefined expression could be the quintic polynomial: t(n) = (17n5 - 320n4 + 2415n3 - 9400n2 + 19168n - 15720)/120 where n = 1, 2, 3, ...
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