0. Limits &
Limits &
Limits at
Limits at
Infinite
Infinite
Infinity
Infinity
Infinity
Infinity
Infinite
Infinite
Limits at
Limits at
Limits &
Limits &

1. Infinite
Limits at
Limits &
Infinity

2. Infinite Limits vs. Limits at Infinity
Recall the limit notation:
limx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,
If we allow N to be  and –,
e.g., limxc f(x) =  and limxc 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.
limxc f(x) = 
limx f(x) c c
limx– f(x)
limxc f(x) = –

3. 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 = ____.

4. 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:

5. 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).

6. Limits at Infinity Q: What is k/ where k is any constant? A: ___
Examples:
Q: What is /k where k is positive constant?
Q: What is /?
A: It’s one of those ____________ forms, i.e, a __________
Examples:
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.

7. 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:

8. 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., 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.