find the limits as ( x \to infty ) and as ( x \to -infty ). (if the limit is infinite, enter ( infty ) or (…

find the limits as ( x \to infty ) and as ( x \to -infty ). (if the limit is infinite, enter ( infty ) or ( -infty ), as appropriate. if the limit does not otherwise exist, enter dne.)\n( y=(3 - x)(1 + x)^{2}(1 - x)^{4} )\n( lim_{x \to infty} f(x)=)\n( lim_{x \to -infty} f(x)=)\nuse this information, together with intercepts, to give a rough sketch of the graph as in this example.

find the limits as ( x \to infty ) and as ( x \to -infty ). (if the limit is infinite, enter ( infty ) or ( -infty ), as appropriate. if the limit does not otherwise exist, enter dne.)\n( y=(3 - x)(1 + x)^{2}(1 - x)^{4} )\n( lim_{x \to infty} f(x)=)\n( lim_{x \to -infty} f(x)=)\nuse this information, together with intercepts, to give a rough sketch of the graph as in this example.

Answer

Explanation:

Step1: Determine the leading term

First, expand the polynomial (y=(3 - x)(1 + x)^{2}(1 - x)^{4}). The leading term of ((3 - x)) is (-x), the leading term of ((1 + x)^{2}) is (x^{2}), and the leading term of ((1 - x)^{4}) is (x^{4}). Multiply the leading terms: ((-x)\times x^{2}\times x^{4}=-x^{7}).

Step2: Find (\lim_{x\rightarrow\infty}f(x))

As (x\rightarrow\infty), consider the function (y = -x^{7}). When (x\rightarrow\infty), (y=-x^{7}\rightarrow-\infty). So (\lim_{x\rightarrow\infty}f(x)=-\infty).

Step3: Find (\lim_{x\rightarrow-\infty}f(x))

As (x\rightarrow-\infty), consider the function (y=-x^{7}). Let (t=-x), when (x\rightarrow-\infty), (t\rightarrow\infty). Then (y = -(-t)^{7}=t^{7}). So (\lim_{x\rightarrow-\infty}f(x)=\infty).

Answer:

(\lim_{x\rightarrow\infty}f(x)=-\infty) (\lim_{x\rightarrow-\infty}f(x)=\infty)