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Question
The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",", if x ≠ 0),(k",", if x = 0):}` is continuous at x = 0 is ______.
Options
0
–1
1
2
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Solution
The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",", if x ≠ 0),(k",", if x = 0):}` is continuous at x = 0 is 1.
Explanation:
The function f is continuous at x = 0 if `lim_(x -> 0) f(x) = f(0)`
We have f(0) = k and
`lim_(x -> 0) f(x)` = `lim_(x -> 0) (1 - cos)/(8x^2)`
= `lim_(x -> 0) (2sin^2 2x)/(8x^2)`
= `lim_(x -> 0) (sin^2 2x)/(4x^2)`
= `lim_(x -> 0) ((sin2x)/(2x))^2` = 1
Hence, k = 1
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