Question

If m is the number of differentiable points of f(x) =  1/(log|x² – 4|) and the value of n for which lim _(x→0) ((xⁿ – sin(xⁿ))/(x – sinⁿx)) has a non-zero finite value, find the value of (m + n).

Answer 1

We have f(x) (1/(log|x² – 4|))

f is not defined when

(x² – 4) = 0, |x² – 4| = 1

(x² – 4) = 0, (x² – 4) = ±1

x² = 4, x² = 4 ± 1

x = ±2, ± √5 , ± √3

Thus m = 6

Also, = lim _(x→0) ((xⁿ – sin(xⁿ))/(x – sinⁿx))

Its limit exists only when n = 1

Hence, the value of (m + n) is 7