[SOLVED] The Squeeze Theorem

**Mike9182** · August 7th 2009, 09:03 AM

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

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

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

**Danny** · August 7th 2009, 09:09 AM

Originally Posted by Mike9182

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

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

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

Since

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

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

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

then (add 1)

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

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