Question

If f(x) = { √2-x x < 2
{ x^3 + k(x+1) x ≥ 2
determine the value of the constant k for which lim exists.
x->2

Answer 1

Since we want the limit to exist, it must have the same value from the left and from the right.

as x increases toward 2, f(x) decreases toward 0

So, we want f(x) to approach zero from the right. That is,

f(x) = x^3 + kx^2 + kx = 0 when x=2
f(2) = 8 + 4k + 2k = 8+6k = 0
k=-4/3

f(x) = x^3 - 4/3x(x+1) for x>=2