Functional Limit

$Applying\quad L'Hôpital'srule,\\ \\ Ans.={ lim }_{ h\rightarrow 0 }\cfrac { \left( 2h+2 \right) f^{ ' }\left( 2h+2+{ h }^{ 2 } \right) }{ \left( 1-2h \right) f^{ ' }\left( { h-h }^{ 2 }+1 \right) } \\ \quad =\cfrac { 2f^{ ' }\left( 2 \right) }{ f^{ ' }\left( 1 \right) } =3$

Let given limit as "L" and Note That :

$L\quad =\quad \frac { f^{ ' }\left( 2 \right) }{ f^{ ' }\left( 1 \right) } \times \lim _{ h\rightarrow 0 }{ \cfrac { 2h\quad +\quad { h }^{ 2 } }{ h\quad -\quad { h }^{ 2 } } } \\ \\ L\quad =\quad \frac { f^{ ' }\left( 2 \right) }{ f^{ ' }\left( 1 \right) } \times \lim _{ h\rightarrow 0 }{ \cfrac { 2\quad +\quad { h } }{ 1\quad -\quad { h } } } \quad \\ \\ L\quad =\quad 2\times \frac { f^{ ' }\left( 2 \right) }{ f^{ ' }\left( 1 \right) } \quad =\quad 3\quad \quad Q.E.D$ .

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Note: Here I used first principle differentiation rule :

$f^{ ' }\left( a \right) \quad =\quad \lim _{ h\rightarrow 0 }{ \frac { f\left( a+h \right) \quad -\quad f\left( a \right) }{ h } }$ .

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