<s>[INST] How to find $ \lim\limits_{ x\to 100 } \frac { 10-\sqrt { x } }{ x-100 }$ Find $ \lim\limits_{ x\to 100 } \dfrac { 10-\sqrt { x } }{ x-100 }$
(without using a calculator and other machines...?)
 [/INST] Recognizing that $\sqrt{x} \approx 10$ makes the numerator vanish, we may be inspired to use a differential approximation:
$$ \sqrt{x} = 10 + \frac{1}{20}(x - 100) + r(x) (x - 100) $$
where $r(x)$ has the property that $\lim_{x \to 100} r(x) = 0$. Therefore,
$$ \lim_{x \to 100} \frac{10 - \sqrt{x}}{x - 100}
= \lim_{x \to 100} \frac{-\frac{1}{20} (x - 100) - r(x) (x - 100)}{x - 100}
= \lim_{x \to 100} -\frac{1}{20} - r(x) = -\frac{1}{20}$$
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