First slide

Introduction to limits

Question

$If f (x) is derivable at x = 3 and f^{'} (3) = 2, then lim_{h \to 0} \frac{f (3 + h^{2}) - f (3 - h^{2})}{2 h^{2}} equals to$

Moderate

A

0
B

2
C

6
D

Does not exists

Solution

$\begin{array}{l} Given that f (x) is derivable at x = 3 and f^{1} (3) = 2 \\ Consider lim_{h \to 0} \frac{f (3 + h^{2}) - f (3 - h^{2})}{2 h^{2}} \\ = lim_{h \to 0} \frac{f^{'} (3 + h^{2}) 2 h - f^{'} (3 - h^{2}) (- 2 h)}{4 h} (b y L' h o s p i t a l r u l e) \\ = lim_{h \to 0} \frac{f^{'} (3 + h^{2}) + f^{'} (3 - h^{2})}{2} = \frac{2 f^{1} (3)}{2} \\ = 2 \end{array}$

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