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Consider the following rational function
f.

f(x)=(-x^(4)+6x^(2)+4x)/(-3x^(5)+x^(4)-8)
Determine
f 's end behavior.

f(x)rarr pick value
vv as
x rarr-oo.

f(x)rarr pick value
vv as
x rarr oo.

Consider the following rational function $f$ . $\newline$ $f(x)=\frac{-x^{4}+6 x^{2}+4 x}{-3 x^{5}+x^{4}-8}$ $\newline$ Determine $f$ 's end behavior. $\newline$ $f(x) \rightarrow$ pick value $\vee$ as $x \rightarrow-\infty$ . $\newline$ $f(x) \rightarrow$ pick value $\vee$ as $x \rightarrow \infty$ .

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Compare linear and exponential growth

Full solution

Q. Consider the following rational function

f

.

\newline

f(x)=\frac{-x^{4}+6 x^{2}+4 x}{-3 x^{5}+x^{4}-8}

\newline

Determine

f

's end behavior.

\newline

f(x) \rightarrow

pick value

\vee

as

x \rightarrow-\infty

.

\newline

f(x) \rightarrow

pick value

\vee

as

x \rightarrow \infty

.

Identify Leading Terms: Identify the leading terms in the numerator and the denominator of the function $f(x) = \frac{-x^4 + 6x^2 + 4x}{-3x^5 + x^4 - 8}$ .
Simplify Expression: Simplify the expression by focusing on the leading terms to find the end behavior as $x$ approaches infinity and negative infinity.
Analyze End Behavior: Analyze the simplified expression as $x$ approaches infinity ( $x \rightarrow \infty$ ) and negative infinity ( $x \rightarrow -\infty$ ).

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