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Refor to the graph of \( y=f(x) \) to the right to describe the behavior of \( \lim f(x) \). Use \( -\infty \) and \( \infty \) where \( x \rightarrow-2^{+} \) Select the correct chaice below a id fill in any answer boxes in your choice. a. \( \lim _{f(x)}=\propto \) \[ x \rightarrow-2^{\prime} \] b. \( \lim f(x)= \) \( \square \) \[ x \rightarrow-2^{-} \] c. \( \lim _{x \rightarrow \infty} f(x)= \) \( \square \)

Ask by Byrd Long. in the United States
Feb 24,2025

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a. \( \lim_{x \to -2^+} f(x) = \infty \) b. \( \lim_{x \to -2^-} f(x) = -\infty \) c. \( \lim_{x \to \infty} f(x) = 0 \)

Solución

Below is one acceptable set of answers based on the graph showing a vertical asymptote at x = –2 (with the function blowing up to +∞ when x approaches –2 from the right and to –∞ when x approaches from the left) and a horizontal asymptote at y = 0:  a. limₓ→–2⁺ f(x) = ∞  b. limₓ→–2⁻ f(x) = –∞  c. limₓ→∞ f(x) = 0 In other words, as x approaches –2 from the right the function increases without bound, while from the left it decreases without bound; and as x goes to infinity the graph levels off at 0. Any answer set that gives these limits is correct.

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It seems you're asking for a description of the limits based on the behavior of the graph of \( y = f(x) \) as \( x \) approaches -2 from the right and from the left, as well as the limit as \( x \) approaches infinity. Since I can't see the graph, I can guide you on how to fill in the answers based on common scenarios: If, as \( x \) approaches -2 from the right (\( -2^{+} \)), the function \( f(x) \) approaches a specific value \( L \), you would write: a. \( \lim_{x \rightarrow -2^{+}} f(x) = L \) And if, as \( x \) approaches -2 from the left (\( -2^{-} \)), the function approaches the same or a different value, you would fill in: b. \( \lim_{x \rightarrow -2^{-}} f(x) = K \) Where \( K \) could be equal to \( L \) or potentially a different limit, depending on the behavior of the graph. Finally, if as \( x \) approaches infinity the function approaches a horizontal asymptote or some specific value \( M \), you would state: c. \( \lim_{x \rightarrow \infty} f(x) = M \) Make sure to replace \( L \), \( K \), and \( M \) with the actual values indicated by the graph! Have fun with your analysis!

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