First slide

Continuity

Question

$Let f (x + y) = f (x) f (y) and f (x) = 1 + (\sin 2 x) g (x) where g (x) is continuous.$ $Then g^{'} (x) is equal to$

Moderate

A

$f (x) g (0)$
B

$2 g (0)$
C

$2 f (x) g (0)$
D

None of these

Solution

$\begin{array}{l} f (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} \lim_{h \to 0} \frac{f (x) f (h) - f (x)}{h} \\ = f (x) \lim_{h \to 0} \frac{f (h) - 1}{h} \\ = f (x) \lim_{h \to 0} \frac{1 + (\sin 2 h) g (h) - 1}{h} \\ = 2 f (x) \lim_{h \to 0} \frac{\sin 2 h}{2 h} \lim_{h \to 0} g (h) \\ = 2 f (x) g (0) \\ H e n c e (3) i s t h e c o r r e c t a n s w e r . \end{array}$

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