Find k if the function f(x) is continuous at x=4.f(x)=cos pi/x, x>=4 f(x)=2square root x, x<4
Since you've not specified where is placed k, we'll assume that f(x) = k`sqrt(x)`
For a function to be continuous at a point x=a, it must obey the following rules:
1) f(a) must exist
2) lim f(x) is finite
3) lim f(x) = f(a)
Now, we'll determine the value of k, knowing that the function is continuous.
lim cos `pi` /x = lim k`sqrt(x)` = f(4)
lim cos `pi` /x = cos `pi` /4 = sqrt2/2
lim k`sqrt(x)` = ksqrt4 = 2k
f(4) = cos ` ` /4 = sqrt2/2
We'll re-write the 3rd condition of continuity:
sqrt2/2 = 2k => k = sqrt2/4
The value of k for the given function to be continuous is k = sqrt2/4.