Question

Describe The End Behavior Of The Function. F(X)=5x⁴+5x+11

A) Limx→[infinity]F(X)=[infinity];Limx→−[infinity]F(X)=[infinity]
B) Limx→[infinity]F(X)=−[infinity];Limx→−[infinity]F(X)=−[infinity]

Answer 1

The function f(x) = 5x⁴ + 5x + 11 has an end behavior such that it rises to infinity as x approaches both infinity and negative infinity, due to the positive, even-powered leading term.

Step-by-step explanation:

The end behavior of the function f(x) = 5x⁴ + 5x + 11 is determined by its highest degree term. Since the leading term is 5x⁴, which has an even exponent and a positive coefficient, the function rises to infinity (+∞) as x approaches infinity and also rises to infinity as x approaches negative infinity (-∞). This is because even-powered functions with positive leading coefficients open upwards at both ends of the graph.

To confirm this end behavior, we evaluate the limits: As x approaches infinity, lim x→∞ f(x) equals infinity (+∞) because the value of x⁴ overpowers the linear term (5x) and the constant term (11) as x becomes very large. Similarly, as x approaches negative infinity, lim x→-∞ f(x) also equals infinity because the even power of x ensures that f(x) is positive, regardless of whether x is positive or negative.