select the true statement regarding the end behavior of the functions.

as the value of $x$ increases, $f$ and $g$ both approach positive infinity.

as the value of $x$ increases, $f$ is the only function to approach 0.

as the value of $x$ increases, $g$ is the only function to approach 0.

Question

select the true statement regarding the end behavior of the functions.

as the value of $x$ increases, $f$ and $g$ both approach positive infinity.

as the value of $x$ increases, $f$ is the only function to approach 0.

as the value of $x$ increases, $g$ is the only function to approach 0.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "End Behavior", "Exponential Functions" ], "new_concepts": [], "current_concepts": [ "End Behavior", "Exponential Functions", "Function Comparison" ] } ## Analyze the behavior of each function as x increases $$ \begin{aligned} &\text{For } f(x): \text{ As } x \text{ increases } (-2, -1, 0, 1, 2, 3), f(x) \text{ is halved each time } (48, 24, 12, 6, 3, 1.5). \text{ Thus, } \lim_{x \to \infty} f(x) = 0.\ &\text{For } g(x) = 2^x - 5: \text{ As } x \to \infty, 2^x \to \infty, \text{ so } \lim_{x \to \infty} g(x) = \infty.\ &\text{For } h(x): \text{ The graph shows that as } x \text{ increases to the right, the curve goes upwards towards positive infinity. Thus, } \lim_{x \to \infty} h(x) = \infty. \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &\text{Statement 1: "As the value of } x \text{ increases, } f \text{ and } g \text{ both approach positive infinity." } \Rightarrow \text{False (since } f(x) \to 0\text{)}.\ &\text{Statement 2: "As the value of } x \text{ increases, } f \text{ is the only function to approach 0."} \Rightarrow \text{True (since } g(x) \to \infty \text{ and } h(x) \to \infty\text{)}.\ &\text{Statement 3: "As the value of } x \text{ increases, } g \text{ is the only function to approach 0."} \Rightarrow \text{False (since } g(x) \to \infty\text{)}. \end{aligned} $$

(A) As the value of x increases, f and g both approach positive infinity.

(B) As the value of x increases, f is the only function to approach 0.

(C) As the value of x increases, g is the only function to approach 0.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "End Behavior" ] }

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Analyze the behavior of each function as x increases

Evaluate the given statements

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