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3.6.notes.b - MAC 2233/001-006 Business Calculus Fall 2011...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Assistants : Matthew Fleeman, Arbin Rai, Tadesse Zerihun, Vindya Kumari, Junyi Tu, Jing- han Meng Notes on finding limits at infinity and infinite limits On finding infinite limits . Suppose that we have a function h ( x ) = f ( x ) g ( x ) (so that f ( x ) is the numerator (top) and g ( x ) is the denominator (bottom)) and that we need to consider the limit of h ( x ) as x c . As before, there are three cases: (1) if g ( c ) negationslash = 0 then substitution works, and we have a (finite) limit. (2) if g ( c ) = 0 and f ( c ) negationslash = 0 then the limit does not exist, and we say we have an “infinite limit” (and see example below for determining the sign). (3) if g ( c ) = f ( c ) = 0 then more work is needed. Example . Determine if the limit lim x 1 + 2+ x 1 - x exists, and if not, say if it is + or −∞ . Solution. When x = 1, we see that the top is nonzero but the bottom is zero. Hence we have an infinite limit. To see if the limit is + or −∞ (and sometimes it is neither), we note that as x 1 + (that is, x > 1 and
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