QuestionAugust 25, 2025

The function f(x) is given by f(x)= 2x^6+5x^4+6x+1 Which of the following correctly describes the end behavior of f as the input values increase without bound? lim _(xarrow infty )f(x)=-infty lim _(xarrow -infty )f(x)=-infty lim _(xarrow infty )f(x)=infty lim _(xarrow -infty )f(x)=infty

The function f(x) is given by f(x)= 2x^6+5x^4+6x+1 Which of the following correctly describes the end behavior of f as the input values increase without bound? lim _(xarrow infty )f(x)=-infty lim _(xarrow -infty )f(x)=-infty lim _(xarrow infty )f(x)=infty lim _(xarrow -infty )f(x)=infty
The function f(x) is given by f(x)= 2x^6+5x^4+6x+1 Which of the following correctly describes the end behavior of f as the input values increase without bound? lim _(xarrow infty )f(x)=-infty lim _(xarrow -infty )f(x)=-infty lim _(xarrow infty )f(x)=infty lim _(xarrow -infty )f(x)=infty

Solution
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Answer

\lim _{x\rightarrow \infty }f(x)=\infty , \lim _{x\rightarrow -\infty }f(x)=\infty Explanation 1. Identify the Leading Term The leading term of f(x) = 2x^6 + 5x^4 + 6x + 1 is 2x^6. 2. Determine End Behavior as x \to \infty As x \to \infty, the term 2x^6 dominates, and since the coefficient is positive, f(x) \to \infty. 3. Determine End Behavior as x \to -\infty As x \to -\infty, the term 2x^6 still dominates. Since the exponent is even, (-x)^6 = x^6, so f(x) \to \infty.

Explanation

1. Identify the Leading Term<br /> The leading term of $f(x) = 2x^6 + 5x^4 + 6x + 1$ is $2x^6$.<br /><br />2. Determine End Behavior as $x \to \infty$<br /> As $x \to \infty$, the term $2x^6$ dominates, and since the coefficient is positive, $f(x) \to \infty$.<br /><br />3. Determine End Behavior as $x \to -\infty$<br /> As $x \to -\infty$, the term $2x^6$ still dominates. Since the exponent is even, $(-x)^6 = x^6$, so $f(x) \to \infty$.
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