First slide

Introduction to limits

Question

If $\lim_{x \to 0} (x^{- 3} \sin 3 x + a x^{- 2} + b)$ exists and is equal to zero, then the value of a + 2b =

Moderate

A

3
B

4
C

0
D

6

Solution

$\lim_{x \to 0} \frac{\sin 3 x + a x + b x^{3}}{x^{3}} (\frac{0}{0} f o r m b y l' H o s p i t a l r u l e) \lim_{x \to 0} \frac{3 \cos 3 x + a + 3 b x^{2}}{3 x^{2}} (D r . t e n d s t o 0, N r . t e n d s t o 0) 3 + a = 0 \Rightarrow a = - 3 \lim_{x \to 0} \frac{- 9 \sin 3 x + 6 b x}{6 x} = - \frac{9}{2} + b = 0 \Rightarrow b = \frac{9}{2}$

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