Question

Let f(x) = `{{:((1 - cos 4x)/x^2",",  "if"  x < 0),("a"",",  "if"  x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if"  x > 0):}`. For what value of a, f is continuous at x = 0?

Answer 1

Here f(0) = a Left hand limit of f at 0 is

`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (1 - cos 4x)/x^2`

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

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

= 8(1)²

= 8.

And right hand limit of f at 0 is

`lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) sqrt(x)/(sqrt(16 + sqrt(x)) - 4)`

= `lim_(x - 0^+) (sqrt(x)(sqrt(16 + sqrt(x)) + 4))/((sqrt(16 + sqrt(x)) + 4)(sqrt(16 + sqrt(x)) - 4))`

= `lim+_(x -> 0^+) (sqrt(x)(sqrt(16 + sqrt(x)) + 4))/(16 + sqrt(x)  16)`

= `lim_(x -. 0^+) (sqrt(16 + sqrt(x)) + 4)`

 = 8

Thus, `lim_(x -> 0+) "f"(x) = lim_(x -> 0^+) "f(x)` = 8.

Hence f is continuous at x = 0 only if a = 8