Evaluate
\frac{2\left(3x-1\right)}{9x^{2}-3x+1}
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\frac{2\left(3x-1\right)}{9x^{2}-3x+1}
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\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}-\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9x^{2}-3x+1 and 9x^{2}+3x+1 is \left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right). Multiply \frac{3x-1}{9x^{2}-3x+1} times \frac{9x^{2}+3x+1}{9x^{2}+3x+1}. Multiply \frac{3x+1}{9x^{2}+3x+1} times \frac{9x^{2}-3x+1}{9x^{2}-3x+1}.
\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)-\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
Since \frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} and \frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{27x^{3}+9x^{2}+3x-9x^{2}-3x-1-27x^{3}+9x^{2}-3x-9x^{2}+3x-1}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
Do the multiplications in \left(3x-1\right)\left(9x^{2}+3x+1\right)-\left(3x+1\right)\left(9x^{2}-3x+1\right).
\frac{-2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
Combine like terms in 27x^{3}+9x^{2}+3x-9x^{2}-3x-1-27x^{3}+9x^{2}-3x-9x^{2}+3x-1.
\frac{-2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}
Factor 81x^{4}+9x^{2}+1.
\frac{-2+54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}
Since \frac{-2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} and \frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} have the same denominator, add them by adding their numerators.
\frac{2\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}
Factor the expressions that are not already factored in \frac{-2+54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}.
\frac{2\left(3x-1\right)}{9x^{2}-3x+1}
Cancel out 9x^{2}+3x+1 in both numerator and denominator.
\frac{6x-2}{9x^{2}-3x+1}
Use the distributive property to multiply 2 by 3x-1.
\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}-\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9x^{2}-3x+1 and 9x^{2}+3x+1 is \left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right). Multiply \frac{3x-1}{9x^{2}-3x+1} times \frac{9x^{2}+3x+1}{9x^{2}+3x+1}. Multiply \frac{3x+1}{9x^{2}+3x+1} times \frac{9x^{2}-3x+1}{9x^{2}-3x+1}.
\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)-\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
Since \frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} and \frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{27x^{3}+9x^{2}+3x-9x^{2}-3x-1-27x^{3}+9x^{2}-3x-9x^{2}+3x-1}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
Do the multiplications in \left(3x-1\right)\left(9x^{2}+3x+1\right)-\left(3x+1\right)\left(9x^{2}-3x+1\right).
\frac{-2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{81x^{4}+9x^{2}+1}
Combine like terms in 27x^{3}+9x^{2}+3x-9x^{2}-3x-1-27x^{3}+9x^{2}-3x-9x^{2}+3x-1.
\frac{-2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}
Factor 81x^{4}+9x^{2}+1.
\frac{-2+54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}
Since \frac{-2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} and \frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)} have the same denominator, add them by adding their numerators.
\frac{2\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}
Factor the expressions that are not already factored in \frac{-2+54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}.
\frac{2\left(3x-1\right)}{9x^{2}-3x+1}
Cancel out 9x^{2}+3x+1 in both numerator and denominator.
\frac{6x-2}{9x^{2}-3x+1}
Use the distributive property to multiply 2 by 3x-1.
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Limits
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