Evaluate `lim_(x to 0) ((1+x)^(6)-1)/((1+x)^(2)-1)`.

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

C

Explanation

Given, `underset(xto0)"lim"((1=x)^(6)-1)/((1+x)^(2)-1)=lim_(hto0)(((1+x)^(6)-1)/(x))/(((1+x)^(2)-1)/(x))`[dividing numerator and denominator by x]
`=underset(xto0)"lim"(((1+x)^(6)-1)/((1+x)-1))/(((1+x)^(2)-1)/((1+x)-1))` `[therefore x to 0 rArr 1+x to 1]`
`=(underset(xto0)"lim"((1+x)^(6)-(1)^(6))/((1+x)-1))/(underset(xto0)"lim"((1+x)^(2)-(1)^(2))/((1+x)-1))` `[therefore underset(xtoa)"lim"(f(x))/(g(x))= (underset(xtoa)"lim"f(x))/(underset(xtoa)"lim"g(x))]`
`=(6(1)^(6-1))/(2(1)^(2-1))` `[therefore underset(xtoa)"lim"(x^(n)-a^(n))/(x-a)=na^(n-1)]`
`=(6 xx 1)/(2 xx 1) = 6/2=3`