lim h 0 f ( 2 h + 2 + h 2 ) - f ( 2 ) f ( h - h 2 + 1 ) - f ( 1 ) , given that f ' ( 2 ) = 6 and f ' ( 1 ) = 4 [2003]

Question

$\lim_{h \to 0} \frac{f (2 h + 2 + h^{2}) - f (2)}{f (h - h^{2} + 1) - f (1)},$ given that $f' (2) = 6$ and $f' (1) = 4$ [2003]

Smart Achievers · Accepted Answer

(4)

$\lim_{h \to 0} \frac{f (2 h + 2 + h^{2}) - f (2)}{f (h - h^{2} + 1) - f (1)}$ $[\frac{0}{0} -form]$

$= \lim_{h \to 0} \frac{f' (2 h + 2 + h^{2}) \cdot (2 + 2 h)}{f' (h - h^{2} + 1) \cdot (1 - 2 h)}$ (Using L'Hospital rule)

$= \frac{f' (2) \cdot 2}{f' (1) \cdot 1} = \frac{6 \times 2}{4 \times 1} = 3$

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