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I n f i n i t e L i m i t s a t L i m i t s & I n f i n i t y I n f i n i t e L i m i t s a t L i m i t s & I n f i n i t y I n f i n i t e L i m i t s a t L i m i t s & I n f i n i t y I n f i n i t e L i m i t s a t L i m i t s & I n f i n i t y

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Infinite Limits at Limits & Infinity Infinite Limits at Limits & Infinity Infinite Limits at Limits & Infinity Infinite Limits at Limits & Infinity

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Recall the limit notation: lim x c f(x) = N means as x is approaching c, but remains unequal to c, the corresponding value of f(x) is approaching to N. Now, Infinite Limits vs. Limits at Infinity If we allow N to be and –, e.g., lim x c f(x) = and lim x c f(x) = –, we have infinite limits. If we allow c to be and –, e.g., lim x f(x) and lim x – f(x), we have limits at infinity. c c lim x c f(x) = lim x c f(x) = – lim x f(x) lim x – f(x)

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We know, when k 0, k /0 is undefined. However, in terms of limit, its really either + or –. Of course, depends on whether k is positive or negative and, also, whether 0 is positive or negative If k is positive, then 0 is positive, then k/0 = ____. If k is positive, then 0 is negative, then k/0 = ____. If k is negative, then 0 is positive, then k/0 = ____. If k is negative, then 0 is negative, then k/0 = ____. Infinite Limits

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Definition of Positive Zero and Negative Zero Positive zero (denoted by +0) is defined as a quantity (usually the denominator) is approaching 0 but remains slightly greater than 0 (i.e, positive). Negative zero (denoted by –0) is defined as a quantity (usually the denominator) is approaching 0 but remains slightly less than 0 (i.e, negative). Examples: 1.2.3. 4.5.6. 7.8.9. Infinite Limits (contd)

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Example 10: Example 11: Final Notes: When we do have the denominator approaching 0 (but not the numerator), we should always consider the ___________ limits, i.e., the _____________ limit and the ______________ limit. Recall that, if the right-sided limit is the same as the left-sided limit (including and – ), then the limit is that quantity. Otherwise, the limit doesnt exist (DNE). Infinite Limits (contd)

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Q: What is k/ where k is any constant? A: ___ Examples:1. 2. Q: What is /k where k is positive constant? A: ___ Examples:1. 2. Q: What is / ? A: Its one of those ____________ forms, i.e, a __________ Examples:1.2.3. Note: A limit in an indeterminate form, e.g. [0/0] and [ / ], only means the limit cant be determined by simply the plug-it-in method, but may be determined by other means. Limits at Infinity

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Q: What is / ? A: As mentioned earlier, its an indeterminate form, i.e., and it depends on _________ and the ___________, whichever is larger. Q: Can one be larger or smaller than another ? A: Yes, of course. Here you go: i) < 2 < 3 < 4 < 5 <... That is, if m and n are positive, and m < n, then m < n. Here we say, the magnitude of m is smaller than the magnitude of n. ii) < 2 < 3 < 4 < 5 <... Although here we see an on the left is smaller than an on the right of the inequality above, we say, 2, 3, 4, 5, etc. have the same magnitude. In general, we say the magnitudes of two s are equal if they have the same exponents. Q: So what do you think the following should be? Summary If the numerator has a larger magnitude than the denominator, then its ___; If the numerator has a smaller magnitude than the denominator, then its ___; If the numerator has the same magnitude than the denominator, then its a _____________, which can be obtained by _____________________. Examples: 1. 2. 3. Limits at Infinity (contd)

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In the previous slide, the meaning of one having a larger magnitude than the other if it has a larger exponent (e.g., 3 > 2, 7 > 4 ) and the meaning of one having the same magnitude as the other if they have the same exponent (e.g., 5 3 and 4 3, 6 7 and 2 7 ). The problem with this definition is that, though it is conceivable that 3 is greater than 2, it is almost unimaginable to say 3 2 has the same magnitude as 2 2, when obviously 3 2 should be greater than 2 2. Therefore, we are going to redo and re-explain the limits a rational function as x or x – (not mentioned previously), as such: When we have to find the limit of a rational function as x, or x –, which term from the numerator and denominator really matters? Answer: _________________ Examples: 1.2. 3.4. 5.6. Limits at Infinity (contd)

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