How do I find the partial fraction decomposition of #(x^4)/(x^4-1)# ?
By Partial Fraction Decomposition, we can write
Let us look at some details.
By rewriting a bit,
Let us find the partial fractions of
by factoring out the denominator,
by splitting into the partial fraction form,
by taking the common denominator,
by simplifying the numerator,
Since the numerator is originally 1, by matching the coefficients,
By adding (1) and (3),
By adding (2) and (4),
By adding (5) and (6),
By plugging (7) into (5),
By plugging (7) and (8) into (1),
By plugging (7) and (8) into (2),
By (5), (6), (9), and (10),
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To find the partial fraction decomposition of (\frac{x^4}{x^4-1}), first factor the denominator as ((x^2-1)(x^2+1)). Then, express (\frac{x^4}{x^4-1}) as (\frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+1}), where (A), (B), (C), and (D) are constants. Then solve for (A), (B), (C), and (D).
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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