Question

Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`

Answer 1

It is given that the function f is continuous at x = 0.

Therefore, `lim_(x -> 0) "f"(x)` = f(0)

⇒ `lim_(x -> 0) (1 - cos4x)/(8x^2)` = k

⇒ `lim_(x -> 0) (2sin^2 - 2x)/(8x^2)` = k

⇒ `lim_(x -> 0) ((sin 2x)/2x)^2` = k

⇒ k = 1

Thus, f is continuous at x = 0 if k = 1.