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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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. $
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