# User:Mozo/plans

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**Theorem** – weak L'Hôpital's rule – Let ⊆ **R**, *u* ∈ , and *f*, *g*: \ {*u*} → **R** be differentiable functions such that ∃lim_{u} g' ∈ **R**, and lim_{u} g' ≠ 0. If

- and

then lim_{u}(f/g) also exists, and

*Proof.* 1) *u* is an accumulation point of he domain of *g*. Since, lim_{u} g' ≠ 0.

2) Both *f* and *g* extend to an → **R** differentiable function, by setting *f*(*u*)=0 and *g*(*u*)=0. Indeed, by Lagrange Theorem, for every *x* ∈ there is an x'∈ between *x* and *u* such that

Hence,