Question

the function ( f ) is given by ( f(x) = 5x^6 - 2x^3 - 3 ). which of the following describes the end behavior of ( f )?
option a: ( limlimits_{x \to -infty} f(x) = -infty ) and ( limlimits_{x \to infty} f(x) = -infty )
option b: ( limlimits_{x \to -infty} f(x) = infty ) and ( limlimits_{x \to infty} f(x) = infty )
option c: ( limlimits_{x \to -infty} f(x) = -infty ) and ( limlimits_{x \to infty} f(x) = infty )
option d: ( limlimits_{x \to -infty} f(x) = infty ) and ( limlimits_{x \to infty} f(x) = -infty )

Explanation:

Step1: Identify Leading Term

The function is \( f(x) = 5x^6 - 2x^3 - 3 \). The leading term (term with highest degree) is \( 5x^6 \).

Step2: Analyze Degree and Leading Coefficient

• Degree of \( x^6 \) is even (6 is even).
• Leading coefficient is \( 5 \), which is positive.

For a polynomial with even degree and positive leading coefficient:

• As \( x \to \infty \), \( x^6 \to \infty \), so \( 5x^6 \to \infty \), hence \( \lim_{x \to \infty} f(x) = \infty \).
• As \( x \to -\infty \), \( (-x)^6 = x^6 \) (since even power), so \( x^6 \to \infty \), and \( 5x^6 \to \infty \), hence \( \lim_{x \to -\infty} f(x) = \infty \).

Answer:

B. \( \lim\limits_{x \to -\infty} f(x) = \infty \) and \( \lim\limits_{x \to \infty} f(x) = \infty \)