Question

Let ƒ(x) = sin x , g(x) = [x + 1] and g{ƒ(x)} = h(x) ; where, [ ] is the greatest integer function. Then h¢$\lim_{x \rightarrow 0}\frac{f(x)}{x^{2}}$is –

• A. Non-existent
• B. 1
• C. –1
• D. None of these

Answer 1

Answer: (A)

Non-existent

Answer 2

h(x) = g{ƒ(x)} = [ƒ(x) + 1] = [sin x + 1]

LHD = h¢$\left( \frac { \pi } { 2 } - 0 \right)$

Ž LHD =

Ž LHD = = = + 

RHD = h¢$\left( \frac { \pi } { 2 } + 0 \right)$

Ž RHD = $\frac { \left[ \sin \left( \frac { \pi } { 2 } + \mathrm { h } \right) + 1 \right] - \left[ \sin \frac { \pi } { 2 } + 1 \right] } { \mathrm { h } }$

Ž RHD = = = –.