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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):}`
Solution
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.
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