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L'Hospitals Rule for Division by Zero
L'HOSPITAL'S RULE FOR 0/0 Suppose that in some deleted neighborhood of a real number c, f'(x) and g'(x) exist and g'(x) ≠ 0. Assume that lim_{x→c} f(x) = 0, lim_{x→c} g(x) = 0. If
exists or is infinite, then (See Figure 5.2.2.) Figure 5.2.2: L'Hospital's Rule Usually the limit will be given by and in this case the proof is very simple.
PROOF IN THE CASE Let Δx be a nonzero infinitesimal. Then f(c) = 0, g(c) = 0, and Taking standard parts we get Intuitively, for x ≈ c the graphs of f(x) and g(x) are almost straight lines of slopes f'(c), g'(c) passing through zero, so the graph of f(x)/g(x) is almost the horizontal line through f'(c)/g'(c) (Figure 5.2.3). Figure 5.2.3 The equation is not always true. For example, g'(c) might be zero or undefined. is sometimes another limit of type 0/0, that is, lim_{x→c} f'(x) = 0 and lim_{x→c} g'(x) = 0. When this happens, l'Hospital's Rule can often be reapplied to lim_{x→c} f'(x)/g'(x). The proof of l'Hospital's Rule in general is fairly long and uses the Mean Value Theorem. It will not be given here. Here are some examples showing how the rule can be applied.


Home Limits, Analytic Geometry, and Approximations L'Hospitals Rule L'Hospitals Rule for 0/0 