Thread: [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 $\displaystyle 1\leq|x-1|\sin^2 x+1 \leq |x-1|+1$

Originally Posted by Mike9182

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.
$