Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.8 The Mean Value Theorem and l’Hôpital’s Rule, #34 · Calculus · Applications of derivatives

It is possible that … has no limit but

It is possible that $f^{'} (x) / g^{'} (x)$ has no limit but $f (x) / g (x) \to L$ . This is why l'Hôpital included an "if."
(a) Find $L$ as $x \to 0$ when $f (x) = x^{2} cos (1/ x)$ and $g (x) = x$ . Remember that cosines are below 1.
(b) From the formula $f^{'} (x) = sin (1/ x) + 2 x cos (1/ x)$ show that $f^{'} / g^{'}$ has no limit as $x \to 0$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.8 The Mean Value Theorem and l’Hôpital’s Rule, #34

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