Number of integral values of `lambda` for which `(lim)_(xvec1)sec^(-1)((lambda^2)/((log)_e x)-(lambda^2)/(x-1))` does not exist is a. 1 b. 2 c. `3` d. 4

Options

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

Answer

C

Explanation

`L=underset(xrarr1)(lim)sec^(-1)((lambda^(2))/(log_(e)x)-(lambda^(2))/(x-1))`
let `x-1=t`
`therefore" "L=underset(trarr0)(lim)sec^(-1)((lambda^(2))/(ln(t+1))-(lambda^(2))/(t))`
`" "=underset(trarr0)(lim)sec^(-1)(lambda^(2)((t-ln(t+1))/(t ln(t+1))))`
`" "=underset(trarr0)(lim)sec^(-1)(lambda^(2)((t-(t-(t^(2))/(2)+...))/(t ln (t+1))))`
`" "=underset(trarr0)(lim)sec^(-1)(lambda^(2)((((t^(2))/(2)-(t^(3))/(3)...))/(t ln(t+1))))`
`" "=underset(trarr0)(lim)sec^(-1)(lambda^(2)((((t^(2))/(2)-(t^(3))/(3)...))/(ln(t+1))))`
`" "=underset(trarr0)(lim)sec^(-1)((lambda^(2))/(2))"exists"``rArr(lambda^(2))/(2)ge1rArr |lamda|gesqrt2`
Thus, L does not exist for `lambda=0, pm1.`