determine the end behavior (*pick two)\n2. ( f(x)=x(x + 3)^{2}(x - 4)^{4} )\n a. ( lim _{x…

determine the end behavior (*pick two)\n2. ( f(x)=x(x + 3)^{2}(x - 4)^{4} )\n a. ( lim _{x \rightarrow-infty} f(x)=-infty )\n b. ( lim _{x \rightarrow-infty} f(x)=infty )\n c. ( lim _{x \rightarrow infty} f(x)=-infty )\n d. ( lim _{x \rightarrow infty} f(x)=infty )\n3. ( f(x)=-(x - 5)(x + 1)^{2}(x + 6)^{3} )\n a. ( lim _{x \rightarrow-infty} f(x)=-infty )\n b. ( lim _{x \rightarrow-infty} f(x)=infty )\n c. ( lim _{x \rightarrow infty} f(x)=-infty )\n d. ( lim _{x \rightarrow infty} f(x)=infty )

determine the end behavior (*pick two)\n2. ( f(x)=x(x + 3)^{2}(x - 4)^{4} )\n a. ( lim _{x \rightarrow-infty} f(x)=-infty )\n b. ( lim _{x \rightarrow-infty} f(x)=infty )\n c. ( lim _{x \rightarrow infty} f(x)=-infty )\n d. ( lim _{x \rightarrow infty} f(x)=infty )\n3. ( f(x)=-(x - 5)(x + 1)^{2}(x + 6)^{3} )\n a. ( lim _{x \rightarrow-infty} f(x)=-infty )\n b. ( lim _{x \rightarrow-infty} f(x)=infty )\n c. ( lim _{x \rightarrow infty} f(x)=-infty )\n d. ( lim _{x \rightarrow infty} f(x)=infty )

Answer

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).