Limits & Limits & Limits at Limits at Infinite Infinite Infinity Infinity Infinity Infinity Infinite Infinite Limits at Limits at Limits & Limits &
Infinite Limits at Limits & Infinity
Infinite Limits vs. Limits at Infinity Recall the limit notation: limxc 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, If we allow N to be and –, e.g., limxc f(x) = and limxc f(x) = –, we have infinite limits. If we allow c to be and –, e.g., limx f(x) and limx – f(x), we have limits at infinity. limxc f(x) = limx f(x) c c limx– f(x) limxc f(x) = –
Infinite Limits We know, when k 0, k /0 is undefined. However, in terms of limit, it’s 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 (cont’d) 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 (cont’d) 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 doesn’t exist (DNE).
Limits at Infinity Q: What is k/ where k is any constant? A: ___ Examples: 1. 2. Q: What is /k where k is positive constant? Q: What is /? A: It’s 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 can’t be determined by simply the “plug-it-in” method, but may be determined by other means.
Limits at Infinity (cont’d) Q: What is /? A: As mentioned earlier, it’s 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 it’s ___; If the numerator has a smaller magnitude than the denominator , then it’s ___; If the numerator has the same magnitude than the denominator , then it’s a _____________, which can be obtained by _____________________. Examples: 1. 2. 3.
Limits at Infinity (cont’d) 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., 53 and 43, 67 and 27). The problem with this definition is that, though it is conceivable that 3 is greater than 2, it is almost unimaginable to say 32 has the same magnitude as 22, when obviously 32 should be greater than 22. 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.