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Evaluate lim(xto1) ((2)/(1-x^(2))-(1)/(...

Evaluate `lim_(xto1) ((2)/(1-x^(2))-(1)/(1-x)).`

Text Solution

Verified by Experts

We have
`underset(xto1)lim((2)/(1-x^(2))+(1)/(x+1))" "(oo-oo"from")`
`=underset(xto1)lim((2)/(1-x^(2))-(1)/(1-x))`
When x=1, the expression `(2)/(1-x^(2))-(1)/(1-x)` assumes the form `oo-oo.` So, we need some simplifieation to express it in the form 0/0. Then,
`underset(xto1)lim((2)/(1-x^(2))-(1)/(1-x))=underset(xto1)lim(2-(1+x))/(1-x^(2))=underset(xto1)lim(1-x)/(1-x^(2))=underset(xto1)lim(1)/(1+x)=1/2`
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