121616949-math.81

# 121616949-math.81 - 4.3 A hard limit 67 This isolates the...

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4.3 A hard limit 67 This isolates the difficult bits in the two limits lim Δ x 0 cos Δ x - 1 Δ x and lim Δ x 0 sin Δ x Δ x . Here we get a little lucky: it turns out that once we know the second limit the first is quite easy. The second is quite tricky, however. Indeed, it is the hardest limit we will actually compute, and we devote a section to it. We want to compute this limit: lim Δ x 0 sin Δ x Δ x . Equivalently, to make the notation a bit simpler, we can compute lim x 0 sin x x . In the original context we need to keep x and Δ x separate, but here it doesn’t hurt to rename Δ x to something more convenient. To do this we need to be quite clever, and to employ some indirect reasoning. The indirect reasoning is embodied in a theorem, frequently called the squeeze theorem. THEOREM 4.1 Squeeze Theorem Suppose that g ( x ) f ( x ) h ( x ) for all x close to a but not equal to a . If lim x a g ( x ) = L = lim x a h ( x ), then lim x a f ( x ) = L .
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