First slide

Introduction to limits

Question

Let $f : R \to R$ be such that f(1) = 3 and f' (1) = 6.Then $lim_{x \to 0} {\{\frac{f (1 + x)}{f (1)}\}}^{1 / x}$ equals.

Moderate

A

1
B

$e^{1 / 2}$
C

$e^{2}$
D

$e^{3}$

Solution

We have,
$\begin{array}{l} lim_{x \to 0} {\{\frac{f (1 + x)}{f (1)}\}}^{1 / x} \\ = lim_{x \to 0} {\{1 + \frac{f (1 + x) - f (1)}{f (1)}\}}^{1 / x} \\ = e lim_{x \to 0} \frac{f (1 + x) - f (1)}{x f (1)} \\ = e^{\frac{1}{f (1)} lim_{x \to 0} \frac{f (x + i) - f (1)}{x}} \\ = e^{\frac{f^{'} (1)}{f (1)}} = e^{6 / 3} = e^{2} [∵ lim_{x \to 0} \frac{f (1 + x) - f (1)}{x} = f^{'} (1)] \end{array}$

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