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Find (lim)(x->1)f(x),w h e r ef(x)=[x^2-...

Find `(lim)_(x->1)f(x),w h e r ef(x)=[x^2-1, xlt=1-x^2-1, x >1`

Text Solution

Verified by Experts

Here `f(x)={{:(x^(2)-1,xle1),(-x^(2)-1,xgt1):}`
at x=1
LHL=`underset(xrarr1^(-))"lim"f(x)`
`=underset(hrarr0)"lim"f(1-h)`
`=underset(hrarr0)"lim"(1-h)^(2)-1`
`=(1-0)^(2)-1=0`
RHL `=underset(xrarr1^(+))"lim"f(x)`
`=underset(hrarr0)"lim"-(1+h)^(2)-1`
`=-(1+0)^(2)-1=-2`
`because` LHL `ne` RHL
`therefore underset(xrarr1)"lim"f(x)` does not exist
Let `1+h=x`
`rArr 1+hrarr1`
`rArr hrarr0`
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