The function $f (x) = \{\begin{cases} 1 - 2 x + 3 x^{2} - 4 x^{3} + ........, |x| < 1 \\ 1, x = - 1 \end{cases} i s$

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Continuous at x=-1

Neither continuous nor derivable at x= -1

Derivable x= -1

None of the above

answer is B.

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Detailed Solution

Since $f (x) = 1 - 2 x + 3 x^{2} - 4 x^{3} + ......... \infty = \frac{1}{{(1 + x)}^{2}}$

$A t x = - 1$

$\lim_{x \to - 1} f (x) = \lim_{x \to - 1} \frac{1}{{(1 + x)}^{2}} = \lim_{x \to 0} \frac{1}{{(1 - 1 + h)}^{2}} = \infty$

But $f (- 1) = 1$

$∴ f (x) i s n o t c o n t r i b u t e s a t x = - 1$

$T h i s i m p l i e s f (x) i s n o t d e r i v a b l e a t x = - 1$

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