how do you find the slant asymtote of f(x) = (x^3)/(x^2-1) ?
To find the slant asymptote of a rational function like f(x) = (x^3)/(x^2-1), you need to perform a polynomial long division between the numerator and the denominator of the function. Here's how to do it:
1. Set up the long division by dividing the numerator (x^3) by the denominator (x^2-1). Write the divisor (x^2-1) on the left side and the dividend (x^3) on the right side. Leave room for the quotient.
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x^2 - 1 | x^3 + 0x^2 + 0x + 0
2. Divide the first term of the dividend (x^3) by the first term of the divisor (x^2) to get x. Write this quotient above the line in the correct position.
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x^2 - 1 | x^3 + 0x^2 + 0x + 0
+ x
3. Multiply the divisor (x^2-1) by the quotient term (x) and write the result below the dividend, aligning similar terms.
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x^2 - 1 | x^3 + 0x^2 + 0x + 0
+ x^3 - x
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- x + 0x + 0
4. Subtract the result from step 3 from the dividend. Write the subtraction result below the line.
________
x^2 - 1 | x^3 + 0x^2 + 0x + 0
+ x^3 - x
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- x + 0x + 0
- x
5. Repeat steps 2-4 until there are no more terms left in the dividend.
In this case, the division stops here because the next term is zero.
6. The resulting quotient is x - 1, and the remainder is -x.
The slant asymptote of f(x) is the quotient (x - 1). So, the slant asymptote of f(x) = (x^3)/(x^2-1) is y = x - 1.