MATH 1A: INDETERMINACIES

Here is a classification/table of all possible indeterminacies, and how they are related to LH (L’Hôpital’s Rule).

Type  Indeterminant  Determinant
Quotient  0/0, ∞/∞ (with + or −)  0/∞ = 0, c/∞ = 0,
Apply LH directly  ∞/c = ±∞

(depending on c)

Product 0 · ∞, 0 · (−∞) ∞ · ∞ = ∞,
Reciprocate before applying LH  0 · 0 = 0,

c · ∞ = ±∞

Difference ∞ − ∞ −∞ − ∞ = −∞,
Rewrite and apply PST (e.g.  ∞ − c = ∞
common denominator, factoring  −∞ + c = −∞, etc.

highest powers, rationalizing, etc.); only then apply LH

Power 0^{0}, ∞^{0}, 1^{∞} 0^{1} = 0, 1^{0} = 1,
Add e^{ln}, then bring the inside  ∞^{1} = ∞, 1^{1} = 1,
power in front of the ln, bring the limit 0^{+∞} = 0, 3^{∞} = ∞,
upstairs into the power, and solve  3^{−∞} = 0,
a new problem altogether with a product  (1/3)^{∞} = 0,
or quotient indeterminacy; finally,  (1/3)^{−∞} = ∞

substitute the answer into the old problem

To summarize, there are 3 types of beasts:

(1) Indeterminacies: one needs to turn them into a quotient indeterminacy and then apply LH.

(2) Determinacies: apply LLs and your answer is right there, no need for fancy techniques and such. In fact, LH is plain dangerous to apply when LL works!

(3) Dubious cases: 1/0 (don’t know), need more info about whether it is 0^{+} or 0^{−}; ∞/0 (don’t know), need more info about 0^{+} or 0^{−}; neither LLs nor LH works in dubious cases. One needs to find more detailed information about the involved zero so as to turn the dubious case into a determinacy andthen find the answer directly.

LH alone doesn’t solve the problems, and in fact, in tricky problems one has to alternate between LLs, LH, and simplification or other PSTs. There is no general rule which goes when, but as a ”rule” of thumb: alternate between LL, LH, simplification, LL, LH, simplification. The end should be with LLs.

Date: March 17, 2004.