Question

Let `[x^y]` denotes the greatest integer of x^r and |x| denotes the modulus of x. Then `lim_(x -> 0) (sum_(r = 1)^(100) [x^r])/(1 + |x|)`

Options

• does not exist

• is –1

• is 1

• is 100

Answer 1

does not exist

Explanation:

R.H.L = `lim_(x -> 0^+) ([x] + [x^2] + [x^3] .... + [x^100]-)/(1 + x)`

= `((-1) + 0 + (-1) + .... + (0))/(1 - 0)` = – 50

∴ L.H.L ≠ R.H.L

∴ Limit does not exist.