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Unformatted text preview: nd closer to 5
4x3 0.00000025 −0.00000000125
0.00000000125 In other words, as x → ±∞, f (x) ≈ 4x3 (1 − 0 + 0) = 4x3 , which is the leading term of f . The
formal proof of Theorem 3.2 works in much the same way. Factoring out the leading term leaves
f (x) = an xn 1 + an−1
+ ... +
an xn As x → ±∞, any term with an x in the denominator becomes closer and closer to 0, and we have
f (x) ≈ an xn . Geometrically, Theorem 3.2 says that if we graph y = f (x), say, using a graphing
calculator, and continue to ‘zoom out,’ the graph of it and its leading term become indistinguishable.
Below are the graphs of y = 4x3 − x + 5 (the thicker line) and y = 4x3 (the thinner line) in two
diﬀerent windows. A view ‘close’ to the origin. A ‘zoomed out’ view. Let’s return to the function in Example 3.1.5, f (x) = x3 (x − 3)2 (x +2) x2 + 1 , whose sign diagram
and graph are reproduced below for reference. Theorem 3.2 tells us that the end behavior is the
same as that of its leading term, x8 . This tells us that the gr...
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