121616949-math.41 - 2.3 Limits 27 as substituting zero for...

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2.3 Limits 27 as “substituting zero for Δ x ,” as that would give - 19 . 6 · 0 - 4 . 9 · 0 0 , which is meaningless. The quantity we are really interested in does not make sense “at zero,” and this is why the answer to the original problem (finding a speed or a slope) was not immediately obvious. In other words, we are generally going to want to figure out what a quantity “approaches” in situations where we can’t merely plug in a value. If you would like to think about a hard example (which we will analyze later) consider what happens to (sin x ) /x as x approaches zero. EXAMPLE 2.2 Does x approach 1.41 as x approaches 2? In this case it is possible to compute the actual value 2 to a high precision to answer the question. But since in general we won’t be able to do that, let’s not. We might start by computing x for values of x close to 2, as we did in the previous sections. Here are some values: 2 . 05 = 1 . 431782106, 2 . 04 = 1 . 428285686, 2 . 03 = 1 . 424780685, 2 . 02 = 1 . 421267040, 2 . 01 = 1 . 417744688, 2 . 005 = 1 . 415980226, 2 . 004 = 1 . 415627070, 2 . 003 = 1 . 415273825, 2 .
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