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I'm slightly confused about one aspect of the conditions for applying L'Hopital's rule.

N.b. apologies in advance for the lack of LaTeX.

L'Hopitals rule:

Let f,g:(a,b) → R be differentiable and let c ε (a,b) be such that f(c)=g(c)=0 and g'(x)≠0 for x≠c.

Then lim(x→ c)[f(x)/g(x)] = lim(x→ c)[f'(x)/g'(x)], provided latter limit exists.

I have lim(x→ 2)[(x²+x-6)/(x²-x-2)] = lim(x→ 2)[(2x+1)/(2x-1)]

I compute this to be 5/3 (which is correct) by L'Hopital's rule.

My point is that this function satisfies f(2)=g(2)=0, but does NOT satisfy g'(x)≠0 for x≠2, as the solution of g'(x)=2x-1=0 is x=0.5; i.e. g'(x)=0 for x=0.5, and 0.5≠c(=2).

So why are we able to apply L'Hopital in this case?

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# L'Hopitals Rule Conditions

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