(Section 2.3: Limits and Infinity I) 2.3.18 Solution Method 2 (The “Short Cut”: Dominant Term Substitution)Again, Nx()=4x3+x±1, and Dx=5x3±2x. Then, fx=. In our limit analysis, we may replace with its dominant term, 4x3, and we may replace with its dominant term, 5x3. This is justified by the Factoring Principle of Dominance (which also gives rise to a rigorous method, but Solution 1 is probably easier): limx±²4x3+x³15x3³2x=limx4x31+14x2³14x3´µ¶·¸¹±1³³´³³5x31³25x2´µ¶·¸¹
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