Find the value of K such that the map f is continuous at x = -2 if f(x) = { K/((x^2)+4) if -5 <= x < -2, (x^3)+7 if -2 <= x <= 4.
To find the value of K such that the map f is continuous at x = -2, we need to ensure that the left-hand limit is equal to the right-hand limit at x = -2.
First, let's find the left-hand limit (LHL) of f(x) as x approaches -2. For -5 <= x < -2, f(x) is given by K/((x^2)+4), so we substitute -2 into the function:
LHL = lim(x -> -2-) f(x) = lim(x -> -2-) [K/((x^2)+4)].
Next, let's find the right-hand limit (RHL) of f(x) as x approaches -2. For -2 <= x <= 4, f(x) is given by (x^3)+7, so we substitute -2 into the function:
RHL = lim(x -> -2+) f(x) = lim(x -> -2+) [(x^3)+7].
To ensure f is continuous at x = -2, we need the LHL to be equal to the RHL:
lim(x -> -2-) [K/((x^2)+4)] = lim(x -> -2+) [(x^3)+7].
Substituting -2 into the respective functions gives:
lim(x -> -2-) [K/((x^2)+4)] = lim(x -> -2+) [(-2^3)+7],
lim(x -> -2-) [K/(0+4)] = lim(x -> -2+) [-1],
lim(x -> -2-) [K/4] = -1.
To find K, we solve for it:
K/4 = -1,
K = -4.
Therefore, the value of K for which the map f is continuous at x = -2 is K = -4.