Explanation of L'Hopital's Rule
Date: 02/06/2001 at 23:34:26 From: Samuel Ward Subject: L'Hospital's Rule In certain cases, L'Hopital's Rule connects the limit of a quotient (f/g) to the limit of the quotient of the derivatives (f'/g'). This is true when f and g go to 0 or infinity at the point where the limit is taken. I understand how to use this rule, and I somewhat understand the proof, but I still do not understand why this happens. Can you help? Please also try to describe explicitly how to think of the roles of the limit, the derivative, and the quotient.
Date: 02/07/2001 at 20:55:05 From: Doctor Fenton Subject: Re: L'Hospital's Rule Hi Samuel, Thanks for writing to Dr. Math. You've posed a very good question. One way you can think of this is to use the idea of derivative: a function f(x) is differentiable at x=a if f(x) is very close to its tangent line y = f'(a)*(x-a) + f(a) near x = a. Specifically, f(x) = f(a) + f'(a)*(x-a) + E1(x) where E1(x) is an error term which goes to 0 as x goes to a. In fact, E1(x) must approach 0 so fast that E1(x) lim ----- = 0 x->a x-a because E1(x) f(x)-f(a) ----- = --------- - f'(a) x-a x-a and we know from the definition of derivative that this quantity has the limit 0 at a. Similarly, if g is differentiable at x = a, g(x) = g(a) + g'(a)*(x-a) + E2(x) where E2(x) is another error term which goes to 0 as x->a. If you're computing the limit of f(x)/g(x) as x->a and if g(a) is not equal to 0, then as x->a, the numerator becomes indistinguishable from f(a) and the denominator from g(a), so the limit is lim f(x) f(a) x->a ---- = ---- g(x) g(a) If both f(a) and g(a) are 0, then we must use the tangent approximations to say that f(x) f(a) + f'(a)*(x-a) + E1(x) ---- = -------------------------- g(x) g(a) + g'(a)*(x-a) + E2(x) f'(a)*(x-a) + E1(x) = --------------------- g'(a)*(x-a) + E2(x) f'(a) + [E1(x)/(x-a)] = --------------------- g'(a) + [E2(x)/(x-a)] and we have seen that the second term becomes negligible as x->a. In other words, when both function values approach 0 as x->a, the ratio of the function values just reduces to the ratio of the slopes of the tangents, because both functions are very close to their tangent lines. Does this clarify the situation? If you still have questions, please write again. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/
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