1.

Let `g(x)=((x-1)^(n))/(log cos^(m)(x-1)), 0 lt xlt 2,` m and n are integers, `m ne 0, n gt 0`, and let p the left h and derivative of `|x-1| at x=1, "If " lim_(x to 1^(+)) g(x)=p, "then"`A. `n=1, m=1`B. `n=1, m=-1`C. `n=2, m=2`D. `n gt 2, m=n`

Answer» Correct Answer - C
We have
`f(x)=|x-1|={{:(,x-1,x ge 1),(,1-x,x lt 1):}`
`therefore ` p=Left hand derivative of f(x) at x=1
`Rightarrow p=underset(x to 1^(-))lim (f(x)-f(1))/(x-1) =underset(x to 1^(-))lim (1-x-0)/(x-1)=-1`
Now,
`underset(x to 1^(+))lim g(x)=p`
`Rightarrow underset(h to 0)lim g(1+h)=-1`
`Rightarrow underset(h to 0)lim (h^(n))/(m log cos h)=1`
`Rightarrow (1)/(m) underset(h to 0)lim (h^(n-2))/(((tan h)/(h)))=1`
`Rightarrow n=2 and "in that case "=(n)/(m)=1`
`Rightarrow m=n=2`


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