Answer:
The Quotient Q(x) is (x + 7)
The Remainder R(x) is (2x - 64)
Limit as x approaches negative or positive infinity (R(x)/(2x² + 5)) is 0
Explanation:
We want to use long division to determine the remainder and quotient in
(2x³+ 14x² + 7x - 29) / (2x² + 5)
This would be done step by step, as follows:
The denominator is written outside, and the numerator inside
We look for a number the when multiplied by the first expression of the funtion outside, gives the first expression of the funtion inside.
We multiply this function by the function outside, and write the resulting values under and subtract from the function inside.
The result is written, and the process is repeated, until the highest power of the function outside is greater the highest power of the function inside.
NOW, LET'S DO IT!
x times 2x = 2x³, so we choose x.
............... x
................................................
2x² + 5 | 2x³+ 14x² + 7x - 29
..............| 2x³ + 5x
..............| 14x² + 2x - 29
7 times 2x² gives 14x², we choose 7
...............x + 7
................................................
2x² + 5 | 2x³+ 14x² + 7x - 29
..............| 2x³ + 5x
..............| 14x² + 2x - 29
..............| 14x² + 35
..............| 2x - 64
We can't go further as the power of x inside is smaller than the power of x outside.
Therefore
Quotient Q(x) = x + 7
Remainder R(x) = 2x - 64
Limit as x approaches negative or positive infinity (R(x)/(2x² + 5))
= lim(x->±infty) (2x - 64) / (2x² + 5)
= 2lim(x->±infty) x(1 - 32/x) / x²(2 + 5/x²)
= 2(0)
= 0