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Question Number 44085 by rk last updated on 21/Sep/18

lim x→0 [((sin ∣x∣)/x)]

$${lim}\:\mathrm{x}\rightarrow\mathrm{0}\:\left[\frac{\mathrm{sin}\:\mid{x}\mid}{{x}}\right] \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18

graph of sin∣x∣

$${graph}\:{of}\:{sin}\mid{x}\mid \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18

graph of ((sin∣x∣)/x)

$${graph}\:{of}\:\frac{{sin}\mid{x}\mid}{{x}} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18

lim_(x→0+)  [((sin∣x∣)/x)]→1 [.]←greatest integer function  lim_(x→0−)  [((sin∣x∣)/x)]→−1  so left hand side limit→−1  right hand side limit→1  so limit does not exist

$$\underset{{x}\rightarrow\mathrm{0}+} {\mathrm{li}{m}}\:\left[\frac{{sin}\mid{x}\mid}{{x}}\right]\rightarrow\mathrm{1}\:\left[.\right]\leftarrow{greatest}\:{integer}\:{function} \\ $$$$\underset{{x}\rightarrow\mathrm{0}−} {\mathrm{lim}}\:\left[\frac{{sin}\mid{x}\mid}{{x}}\right]\rightarrow−\mathrm{1} \\ $$$${so}\:{left}\:{hand}\:{side}\:{limit}\rightarrow−\mathrm{1} \\ $$$${right}\:{hand}\:{side}\:{limit}\rightarrow\mathrm{1}\:\:{so}\:{limit}\:{does}\:{not}\:{exist} \\ $$

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