CalcNotes0206-page1 - x 2 so that the middle part becomes...

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(Section 2.6: The Squeeze (Sandwich) Theorem) 2.6.1 SECTION 2.6: THE SQUEEZE (SANDWICH) THEOREM PART A: EXAMPLES We will formally state the Squeeze (Sandwich) Theorem later. Example 1 below is one of many basic examples in which we use the Squeeze (Sandwich) Theorem to prove that the limit of a function is 0, where the function includes a trig function. Example 1 Prove that lim x ± 0 x 2 cos 1 x ² ³ ´ µ · = 0 . Idea It makes sense that (something approaching 0) times (something bounded between two values) will approach 0. Solution Show how cos 1 x ± ² ³ ´ µ is bounded. ± 1 ² cos 1 x ³ ´ µ · ¸ ² 1 , if x ± 0 ± Multiply all three parts by
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Unformatted text preview: x 2 so that the middle part becomes the expression we want to take the limit of. If x , then x 2 > , and so we have an appropriate step where none of the inequality symbols have to be reversed. x 2 x 2 cos 1 x x 2 , if x As x , the left and right parts approach 0. Therefore, by the Squeeze (Sandwich) Theorem, the middle part is forced to approach 0, as well. The middle part is squeezed or sandwiched between the left and right parts, and it must approach the same limit that the other two are approaching....
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