# How do I find the partial fraction decomposition of #(2x)/((x+3)(3x+1))# ?

By decomposing into smaller fractions,

by taking the common denominator,

By comparing the numerators,

Hence,

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To find the partial fraction decomposition of ( \frac{2x}{(x+3)(3x+1)} ), you first express the fraction in the form of partial fractions. Here's how you do it:

- Write the fraction as ( \frac{A}{x+3} + \frac{B}{3x+1} ).
- Multiply both sides by the denominator ((x+3)(3x+1)) to get rid of the fractions.
- Solve for ( A ) and ( B ) by equating coefficients of like terms on both sides.

[ \frac{2x}{(x+3)(3x+1)} = \frac{A}{x+3} + \frac{B}{3x+1} ]

Multiply both sides by ((x+3)(3x+1)):

[ 2x = A(3x+1) + B(x+3) ]

Expand the right side and equate coefficients:

[ 2x = 3Ax + A + Bx + 3B ]

Now, equate coefficients of like terms:

For ( x ) terms: ( 2 = 3A + B )

For constant terms: ( 0 = A + 3B )

Now, solve these two equations to find the values of ( A ) and ( B ), which will give you the partial fraction decomposition.

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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.

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