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Thread: [SOLVED] The Squeeze Theorem

  1. #1
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    [SOLVED] The Squeeze Theorem

    I have an explanation of how to solve this but don't understand one of the steps in it:

    $\displaystyle \lim {x\to1} |x-1| \sin^2 x+1$



    I don't understand how it follows that $\displaystyle 1\leq|x-1|\sin^2 x+1 \leq |x-1|+1$
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  2. #2
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    Quote Originally Posted by Mike9182 View Post
    I have an explanation of how to solve this but don't understand one of the steps in it:

    $\displaystyle \lim {x\to1} |x-1| \sin^2 x+1$



    I don't understand how it follows that $\displaystyle 1\leq|x-1|\sin^2 x+1 \leq |x-1|+1$
    Since

    $\displaystyle
    0 \le \sin^2 x \le 1
    $

    then (multiply by $\displaystyle |x-1|$ )


    $\displaystyle
    0 \le |x-1|\sin^2 x \le |x-1|
    $

    then (add 1)

    $\displaystyle
    1 \le |x-1|\sin^2 x + 1 \le |x-1| + 1.
    $
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