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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §2.1 The Derivative of a Function, #25 · Calculus · Derivatives

### *26 Suppose … as

Guess the limits by an informal working rule. Set $Δ t = 0.1$ and $- 0.1$ and imagine $Δ t$ becoming smaller:
(a) $\frac{1 + Δ t}{2 + Δ t}$
(b) $\frac{∣Δ t ∣}{Δ t}$
(c) $\frac{Δ t + ( Δ t ) ^{2}}{Δ t - ( Δ t ) ^{2}}$
(d) $\frac{t + Δ t}{t - Δ t}$

### *26
Suppose $f (x) / x \to 7$ as $x \to 0$ . Deduce that $f (0) = 0$ and $f^{'} (0) = 7$ . Give an example other than $f (x) = 7 x$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §2.1 The Derivative of a Function, #25

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