L'Hospital's rule for ∞/∞
Suppose that for all x in some real open interval (c, b), both f ′(x) and g ′(x) exist and g ′(x) ≠ 0. Assume that
and
.
If limx→ c⁺ (f ′(x) / g ′(x)) exists, then holds L'Hospital's rule
.
Explanation
Let L = limx→ c⁺ (f ′(x) / g ′(x)). By the mean value theorem every real solution of
c < x < y < c + r
is a partial solution of
x < t < y,
which we write as
x < t < y,
.
Let y1 ≈ c and y1 > c. Then f (y1) and g (y1) are positive infinite, and so is their product K = f (y1) g (y1). By the M, δ condition for limx→ c⁺ g ′(x) = ∞, for every real M there is a real δ (M) such that every real solution of
c < x < c + δ (M)
is a solution of g (x) > M. Let δ1 be such that 0 < δ1≤ δ (K) and c + δ1 < y1. Consider any x1 with c < x1 < c + δ1. By the transfer principle, g (x1) > K. Moreover,
c < x1 < y1 < c + x
so by the partial solution theorem there is a t1 such that the formula holds. Then
.
Also
so (f (y1) / g (x1)) ≈ 0. Similarly (g (y1) / g (x1)) ≈ 0. Taking standard parts in the formula, we have
whence
.
Since this holds for c < x1 < c + δ1 we see from the definition of the limit that
.
HistoryThe German-American mathematician Abraham Robinson described this rule of the French mathematician Guillaume de l'Hospital in the early 1960 in this way. |