The Theorem of the Day: L’ Hospital’s Rule

Let $f(x)$ and $g(x)$ be real-valued functions of a real variable $x$. Let $a ∈ R$ and suppose that, in some open interval containing $a$, $f(x$ and $g(x)$ are differentiable except possibly at $x = a$, with $g′(x)$ being nowhere zero. 

Suppose, moreover that

Then

provided this limit exists, satisfies

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