Given P(x)=5x^5−9x^3−9x^2+3x+9,
P(x)→ ? if x→−∞,
P(x)→ ? if x→∞.
the 5th order term is the driver
it is an odd power, so the function will have the same sign as x
ty!
To find the limiting behavior of a polynomial function as x approaches positive or negative infinity, we need to consider the highest power of x in the polynomial.
For the given polynomial function P(x) = 5x^5 - 9x^3 - 9x^2 + 3x + 9, the highest power of x is 5. Therefore, we can determine the limiting behavior by looking at the sign of the term with the highest power as x approaches positive or negative infinity.
1. P(x) → ? as x → -∞ (x approaches negative infinity):
As x approaches negative infinity, the behavior of the polynomial depends on the sign of the term with the highest power, which is 5x^5.
When x → -∞, the sign of the term 5x^5 is determined by the coefficient 5. Since 5 is positive, the term 5x^5 will be positive.
Therefore, as x approaches negative infinity (x → -∞), the polynomial P(x) will also approach positive infinity: P(x) → +∞.
2. P(x) → ? as x → ∞ (x approaches positive infinity):
Similarly, when x approaches positive infinity, the behavior of the polynomial depends on the sign of the term with the highest power, which is again 5x^5.
When x → ∞, the sign of the term 5x^5 is determined by the coefficient 5. Since 5 is positive, the term 5x^5 will be positive.
Therefore, as x approaches positive infinity (x → ∞), the polynomial P(x) will also approach positive infinity: P(x) → +∞.