Question

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.)
( y=(3 - x)(1 + x)^{2}(1 - x)^{4} )
( lim_{x \to infty} f(x)=)
( lim_{x \to -infty} f(x)=)
use this information, together with intercepts, to give a rough sketch of the graph as in this example.

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

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

Step3: Find \(\lim_{x

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

Answer:

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