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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




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 (y1g (y1). By the Mδ 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



so (f (y1) / g (x1)) ≈ 0. Similarly (g (y1) / g (x1)) ≈ 0. Taking standard parts in the formula, we have



Since this holds for c < x1 < c + δ1 we see from the definition of the limit that




The German-American mathematician Abraham Robinson described this rule of the French mathematician Guillaume de l'Hospital in the early 1960 in this way.

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