Question

Let g and h be the functions defined by g(x) = sin(π/2 x) + 4 and h(x) = -1/4 x³ + 3/4 x + 9/2. If f is a function that satisfies g(x) ≤ f(x) ≤ h(x) for -1 < x < 2 what is

lim f(x)?
x → 1

a. 4
b. 9/2
c. 5
d. the limit cannt be determined from the given information

Answer 1

The limit of the function f(x) as x approaches 1 is 5, since it lies between the limits of g(x) and h(x) at the point x = 1, which are 5 and 9/2 respectively.

Step-by-step explanation:

The question asks for the limit of the function f(x) as x approaches 1, given that g(x) ≤ f(x) ≤ h(x) for -1 < x < 2, where g(x) = sin(π/2 x) + 4 and h(x) = -1/4 x³ + 3/4 x + 9/2. To find the limit as x approaches 1, we evaluate g(1) and h(1). The function g(x) at x=1 is g(1) = sin(π/2 * 1) + 4 = sin(π/2) + 4 = 1 + 4 = 5. The function h(x) at x=1 is h(1) = -1/4 * 1³ + 3/4 * 1 + 9/2 = -1/4 + 3/4 + 9/2 = 9/2.

Since f(x) is sandwiched between g(x) and h(x), and both g(1) and h(1) are equal to 5 and 9/2 respectively, the limit of f(x) as x approaches 1 must satisfy 5 ≤ lim f(x) ≤ 9/2. Hence, the only possible value for the limit that meets this condition is 5, making the answer c. 5.