Chapter 1: Limits
Section 1.3: Limit Laws
Use the Squeeze theorem to show limx→0x2 sin1/x=0.
In Example 1.1.8, Principle 1.1.1 was used to establish limx→0x sin1/x=0. Because sin1/x≤1, and x→0, the limit is zero by Principle 1.1.1. The limit given in the present example would also yield to Principle 1.1.1 because x2→0 also.
To establish the given limit via the Squeeze theorem, start with the following bound on sin1/x.
and multiply through by the positive quantity x2. This gives
Since both x2 and −x2 go to zero, by the Squeeze theorem so also must x2sin1/x go to zero.
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