If mathop {Lim}limits_{x to 0} (x^{-3} sin 3x | vedclass

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Question

(A) Given the limit $\mathop {Lim}\limits_{x 	o 0} (\frac{\sin 3x}{x^3} + \frac{a}{x^2} + b) = 0$.Combining the terms: $\mathop {Lim}\limits_{x 	o 0

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$a = -3, b = 9/2$

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(A) Given the limit $\mathop {Lim}\limits_{x 	o 0} (\frac{\sin 3x}{x^3} + \frac{a}{x^2} + b) = 0$.Combining the terms: $\mathop {Lim}\limits_{x 	o 0} \frac{\sin 3x + ax + bx^3}{x^3} = 0$.Using the Taylor series expansion for $\sin 3x = 3x - \frac{(3x)^3}{3!} + \frac{(3x)^5}{5!} - \dots = 3x - \frac{27x^3}{6} + O(x^5)$.Substituting this into the expression: $\mathop {Lim}\limits_{x 	o 0} \frac{3x - \frac{27}{6}x^3 + ax + bx^3}{x^3} = 0$.For the limit to exist,the coefficient of $x$ must be zero: $3 + a = 0 \Rightarrow a = -3$.Now,the expression becomes $\mathop {Lim}\limits_{x 	o 0} \frac{-\frac{27}{6}x^3 + bx^3}{x^3} = -\frac{27}{6} + b = 0$.Solving for $b$: $b = \frac{27}{6} = \frac{9}{2}$.

If $\mathop {Lim}\limits_{x \to 0} (x^{-3} \sin 3x + ax^{-2} + b)$ exists and is equal to zero,then:

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