Question

determine the end behavior (*pick two)

1. ( f(x)=x(x + 3)^{2}(x - 4)^{4} )

a. ( lim _{x
ightarrow-infty} f(x)=-infty )
b. ( lim _{x
ightarrow-infty} f(x)=infty )
c. ( lim _{x
ightarrow infty} f(x)=-infty )
d. ( lim _{x
ightarrow infty} f(x)=infty )

1. ( f(x)=-(x - 5)(x + 1)^{2}(x + 6)^{3} )

a. ( lim _{x
ightarrow-infty} f(x)=-infty )
b. ( lim _{x
ightarrow-infty} f(x)=infty )
c. ( lim _{x
ightarrow infty} f(x)=-infty )
d. ( lim _{x
ightarrow infty} f(x)=infty )

Explanation:

2. \(f(x)=x(x + 3)^{2}(x - 4)^{4}\)

Step1: Determine the leading term

The leading term of \(f(x)\) is obtained by multiplying the leading terms of each factor. The leading term of \(x\) is \(x\), of \((x + 3)^{2}\) is \(x^{2}\), and of \((x - 4)^{4}\) is \(x^{4}\). So the leading term is \(x\times x^{2}\times x^{4}=x^{7}\).

Step2: Analyze the limit as \(x\to-\infty\)

For \(y = x^{n}\), when \(n\) is odd (\(n = 7\) here) and \(x\to-\infty\), \(x^{n}\to-\infty\). So \(\lim_{x\to-\infty}f(x)=-\infty\) (Option a).

Step3: Analyze the limit as \(x\to\infty\)

When \(n\) is odd (\(n = 7\)) and \(x\to\infty\), \(x^{n}\to\infty\). So \(\lim_{x\to\infty}f(x)=\infty\) (Option d).

3. \(f(x)=-(x - 5)(x + 1)^{2}(x + 6)^{3}\)

Step1: Determine the leading term

The leading term of \(-(x - 5)\) is \(-x\), of \((x + 1)^{2}\) is \(x^{2}\), and of \((x + 6)^{3}\) is \(x^{3}\). So the leading term is \(-x\times x^{2}\times x^{3}=-x^{6}\).

Step2: Analyze the limit as \(x\to-\infty\)

For \(y=-x^{n}\), when \(n\) is even (\(n = 6\)) and \(x\to-\infty\), \(x^{n}\to\infty\), so \(-x^{n}\to-\infty\). So \(\lim_{x\to-\infty}f(x)=-\infty\) (Option a).

Step3: Analyze the limit as \(x\to\infty\)

When \(n\) is even (\(n = 6\)) and \(x\to\infty\), \(x^{n}\to\infty\), so \(-x^{n}\to-\infty\). So \(\lim_{x\to\infty}f(x)=-\infty\) (Option c).

Answer:

For \(f(x)=x(x + 3)^{2}(x - 4)^{4}\): a. \(\lim_{x\to-\infty}f(x)=-\infty\), d. \(\lim_{x\to\infty}f(x)=\infty\).
For \(f(x)=-(x - 5)(x + 1)^{2}(x + 6)^{3}\): a. \(\lim_{x\to-\infty}f(x)=-\infty\), c. \(\lim_{x\to\infty}f(x)=-\infty\).