Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.8 The Mean Value Theorem and l’Hôpital’s Rule, #33 · Calculus · Applications of derivatives

Suppose

Suppose $f (0) = 0$ . If $f (x) / x$ has a limit as $x \to 0$ , that limit is better known to us as ______. l'Hôpital's Rule looks instead at the limit of ______.
Conclusion from l'Hôpital: The limit of $f^{'} (x)$ , if it exists, agrees with $f^{''} (0)$ . Thus $f^{'} (x)$ cannot have a "removable ______."

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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, #33

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