The end behavior of the power function is as x tends to infinity, f(x) tends to zero.
In this question,
The power function is [tex]f(x)=3^{-x}2[/tex]
The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.
The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
The graph below shows the behavior of the function f(x).
The above equation has the degree of 1, which is odd and the leading coefficient has the positive coefficient.
Then the end behavior is
As x -> ∞,
⇒ [tex]\lim_{x \to \infty} f(x)[/tex]
⇒ [tex]\lim_{x \to \infty} 3^{-x}2[/tex]
⇒ [tex]2\lim_{x \to \infty} 3^{-x}[/tex]
⇒ [tex]2\lim_{x \to \infty} 3^{-\infty}[/tex]
⇒ 2(0)
⇒ 0
Thus as x → ∞, f(x) → 0.
Hence we can conclude that the end behavior of the power function is as x tends to infinity, f(x) tends to zero.
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