L'Hôpital's Rule

Real Analysis · Axiom Academy

When a limit collapses to 0/0 or ∞/∞, the answer is hiding in the ratio of slopes — replace each function by its derivative. Suppose f and g both head to 0 as , so the quotient f/g becomes the indeterminate form 0/0 . If f and g are differentiable near a with , then the limit is recovered from the ratio of their derivatives . — provided the right-hand limit exists. Geometrically, both curves pass through height 0 at x = a . Near that crossing each function looks like a straight line of slope f'(a) and g'(a) , so the tiny rise of f versus the tiny rise of g is just f'(a)/g'(a) — that is the value the quotient is sneaking up on. Worked example: the classic limit Direct substitution gives 0/0 , so the rule applies. Differentiate top and bottom: The value 1 is exactly the slope of at the origin. L'Hôpital's Rule applies only to 0/0 and . Used on a quotient that already has a value, it lies. For example is plainly by substitution — but differentiating top and bottom gives , the wrong answer. Always verify the indeterminate form before you reach for derivatives. 2. Why It Works: the Mean Value Theorem The rule is not magic — it falls out of Cauchy's Mean Value Theorem . With f(a)=g(a)=0 , the quotient equals , and Cauchy's MVT says that ratio equals the ratio of derivatives at some interior point c between a and x . As , the trapped point c is squeezed toward a as well. So the quotient — which always equals f'(c)/g'(c) — converges to f'(a)/g'(a) :

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