Question

The function f defined by f(x) =$f(x) = \sqrt{\frac{8}{1 + \cos x} + \frac{8}{1 - \cos x}}$ for x ≠ 0 for x = 0 is

• A. Continuous and derivable at x = 0
• B. Neither continuous nor derivable at x = 0
• C. Continuous but not derivable at x = 0
• D. None of these

Answer 1

Answer: (A)

Continuous and derivable at x = 0

Answer 2

The function is continuous at x = 0, because

$\lim _ { x \rightarrow 0 } f ( x ) = \lim _ { x \rightarrow 0 } \frac { \sin x ^ { 2 } } { x } = \lim _ { x \rightarrow 0 } \left( \frac { \sin x ^ { 2 } } { x ^ { 2 } } \right)$. x = 0 = f(0).

Also

= 1

and

$L f ^ { \prime } ( 0 ) = \lim _ { h \rightarrow 0 } \frac { f ( 0 - h ) - f ( 0 ) } { - h } = \lim _ { h \rightarrow 0 } \frac { \frac { \sinh ^ { 2 } } { - h } - 0 } { - h } = \lim _ { h \rightarrow 0 } \frac { \sinh ^ { 2 } } { h ^ { 2 } }$= 1

so that, f(x) is derivable at x = 0