Box "1/0"
More generally, we have the type L/0 here, where L is not zero
(zero over zero is a special type), but L can be infinite. Since we
can always factor out sign, we can assume that
L > 0.
Since we can also always do
L/0 = L⋅(1/0),
then the value of L will not influence the outcome, so it is enough to
know what to do with 1/0.
Assume that we want to find the limit of an expression
g/h at some a, and we know that g goes to
1 while h goes to 0 at a.
Standard procedure: We try to find some reduced neighborhood of a
so that either h > 0 on this neighborhood (then h
is of the type 0+ and we remember that
1/0+ = ∞);
or so that h < 0 on this neighborhood (then
h is of the type 0- and we remember that
1/0- = −∞).
If we are not able to find such a neighborhood and h attains both
signs arbitrarily close to a, then this particular 1/0 leads to a
limit that does not exist. Very often we can get information about signs of
h by considering one-sided limits.
Example:
Find the limit of the function
f (x) = 1/(x − 3)2
at x = 3.
Solution: When we substitute the limit point into f, we see
that we face the type 1/0. What is the sign of the denominator for x
near 3? For x ≠ 3 we always have
(x − 3)2 > 0, so in our case we
have 1/0+ and the answer is
∞.
Example:
Find the limit of the function
f (x) = 1/(ln(x) − 1)
at x = e.
Solution: When we substitute the limit point into f, we see
that we face the type 1/0. What is the sign of the denominator for x
near e? Experience tells us that this depends on what side x is.
If x > e, then
ln(x) > 1,
that is, (ln(x) − 1) > 0. This shows that
the denominator is positive on some right reduced neighborhood of e,
so if we consider limit at e from the right, then the answer is
1/0+ = ∞.
On the other hand, if x < e, then
ln(x) < 1, that is,
(ln(x) − 1) < 0. This shows that
the denominator is negative on some left reduced neighborhood of e,
so if we consider limit at e from the left, then the answer is
1/0- = −∞.
Since the limit from the right of the given expression at e is
different from its limit at e from the left, the conclusion is that
the given limit does not exist.
This type of reasoning is used very often, for more details see the section
Basic properties in Theory -
Limits, of some interest is
this problem
in Solved Problems - Limits.
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