jee-advanced 2003 Q13

jee-advanced · India · screening Differentiation from First Principles Limit Involving Derivative Definition of Composed Functions

13. $\lim _ { ( h \rightarrow 0 ) } \left( \mathrm { f } \left( 2 \mathrm {~h} + 2 + \mathrm { h } ^ { 2 } \right) - \mathrm { f } ( 2 ) \right) / \left( \mathrm { f } \left( \mathrm { h } - \mathrm { h } ^ { 2 } + 1 \right) - \mathrm { f } ( 1 ) \right)$, given that $f ^ { \prime } ( 2 ) = 6$ and $f ^ { \prime } ( 1 ) = 4$ :
(a) does not exists
(b) is equal to $- 3 / 2$
(c) is equal to $3 / 2$
(d) is equal to 3

Let $f ( x )$ be a differentiable function with $f ^ { \prime } ( 1 ) = 4$ and $f ^ { \prime } ( 2 ) = 6$, where $f ^ { \prime } ( c )$ is the derivative of $f ( x )$ at $x = c$. Then the limit of $\frac { f \left( 2 + 2 h + h ^ { 2 } \right) - f ( 2 ) } { f \left( 1 + h - h ^ { 2 } \right) - f ( 1 ) }$, as $h \rightarrow 0$,

13. $\lim _ { ( h \rightarrow 0 ) } \left( \mathrm { f } \left( 2 \mathrm {~h} + 2 + \mathrm { h } ^ { 2 } \right) - \mathrm { f } ( 2 ) \right) / \left( \mathrm { f } \left( \mathrm { h } - \mathrm { h } ^ { 2 } + 1 \right) - \mathrm { f } ( 1 ) \right)$, given that $f ^ { \prime } ( 2 ) = 6$ and $f ^ { \prime } ( 1 ) = 4$ :\\
(a) does not exists\\
(b) is equal to $- 3 / 2$\\
(c) is equal to $3 / 2$\\
(d) is equal to 3\\

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