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Find lim(xrarr0) f(x) where f(x){{:((...

Find `lim_(xrarr0) f(x)` where
`f(x){{:((x)/(|x|),xne0),(0,x=0):}`

Text Solution

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`f(x){{:((x)/(|x|),xne0),(0,x=0):}`
at x=0
LHL `underset(xrarr0^(-))"lim"f(x)`
`=underset(hrarr0)"lim"f(0-h)`
`=underset(hrarr0)(h)/(|-h|)`
`=underset(hrarr0)"lim"(h)/(-h)=underset(hrarr0)"lim"(-1)=-1`
Let `0-h=x`
`rArr 0-hrarr0`
`rArr hrarr0`
RHL`=underset(Xrarr0^(+))"lim"f(x)`
` underset(hrarr0)"lim"f(0+h)`
`=underset(hrarr0)"lim"(h)/(|h |)`
` =underset(hrarr0)"lim"(h)/(h)=underset(hrarr0)(1)=1`
`because LHLneRHL`
`therefore underset(xrarr0)"lim"` f(x) does not exist.
Let `0+h=x`
`rArr 0+hrarr0`
`rArr hrarr0`
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