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Evaluate lim(xrarr0)((1+x)^(6)-1)/((1+x)...

Evaluate `lim_(xrarr0)((1+x)^(6)-1)/((1+x)^(5)-1).`

Text Solution

Verified by Experts

Put `(1+x)=y,` so that when `x to0 then y to1.`
`thereforelim_(xto0)((1+x)^(6)-1)/((1+x)^(5)-1)`
`=lim_(yto1)((y^(6)-1)/(y^(5)-1))=(lim_(yto1)((y^(6)-1)/(y-1)))/(lim_(yto1)((y^(5)-1)/(y-1)))=(lim_(y to1)((y^(6)-1^(6))/(y-1)))/(lim_(y to1)((y^(5)-1^(5))/(y-1)))(6xx1^((6-1)))/(5xx1^((5-1)))`
`=(6xx1^(5))/(5xx1^(4))=6/5" "[becauselim_(xtoa)((x^(n)-a^(n))/(x-a))=na^(n-1)].`
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