How do you find the maclaurin series expansion of #f(x) = (xsin x)/ x^3#?
I don't think it's necessary to determine the limits of f(0), f'(0), f''(0),...,
for the Maclaurin series coefficients for
Given g(x)= x sinx, the Maclaurin series can be found.
Thus, without x = 0.
Ultimately, the Maclaurin series serves as the function's power series.
x = 0.
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To find the Maclaurin series expansion of ( f(x) = \frac{x  \sin(x)}{x^3} ), follow these steps:
 Find the Maclaurin series expansions for ( x ) and ( \sin(x) ).
 Subtract the series expansion of ( \sin(x) ) from the series expansion of ( x ).
 Divide the resulting series by ( x^3 ).
The Maclaurin series expansions are as follows:

( x ): [ x = x ]

( \sin(x) ): [ \sin(x) = x  \frac{x^3}{3!} + \frac{x^5}{5!}  \frac{x^7}{7!} + \cdots ]
Now, subtract the series expansion of ( \sin(x) ) from the series expansion of ( x ):
[ x  \sin(x) = 0 + \frac{x^3}{3!}  \frac{x^5}{5!} + \frac{x^7}{7!}  \cdots ]
Finally, divide the resulting series by ( x^3 ):
[ \frac{x  \sin(x)}{x^3} = \frac{1}{3!}  \frac{x^2}{5!} + \frac{x^4}{7!}  \cdots ]
Thus, the Maclaurin series expansion of ( f(x) = \frac{x  \sin(x)}{x^3} ) is: [ f(x) = \frac{1}{3!}x^{3}  \frac{1}{5!}x^{1} + \frac{1}{7!}x + \cdots ]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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