Problem

Analyze the displayed graph to determine the long-term ordering of \(L\) and \(E\), justify it from their end behavior and rates, and state what exact crossing or earlier ordering the graph does not determine.

Coordinate graph showing a rising line and a decaying exponential curve that falls toward zero, so the line becomes greater as x increases.

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long-term = L(x)>E(x) for every real x; evidence = L→∞ and E→0⁺ as x→∞; rate contrast = linear growth versus decay; not determined = no early-input qualification is needed

long-term = L(x)>E(x) for all sufficiently large x; evidence = L→∞ while E→0⁺ as x→∞; rate contrast = constant positive linear rate versus multiplicative decay; not determined = exact crossing input or earlier dominance without equations or graph coordinates

long-term = E(x)>L(x) for sufficiently large x; evidence = exponentials always win; rate contrast = decay is still exponential; not determined = nothing about crossings

long-term = L(x)>E(x) after x=0; evidence = E approaches asymptote y=0; rate contrast = line increases while E decays; not determined = exact crossing x=0

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Analyze the displayed graph to determine the long-term ordering of \(L\) and \(E\), justify it from their end behavior and rates, and state what exact crossing or earlier ordering the graph does not determine.

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