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Question
Discuss the continuity of the function at the point given. If the function is discontinuous, then remove the discontinuity.
f (x) = `(sin^2 5x)/x^2` for x ≠ 0
= 5 for x = 0, at x = 0
Solution
Given f(0) = 5 ..........(1)
Consider,
`lim_(x ->0) f(x) = lim_(x ->0)(sin^2 5x)/x^2 `
= `lim_(x ->0) ((sin 5x)/(5x) xx 5)^2 (thereforex -> 0, x ≠0)`
= 1 x 5²
= 25
From (i) and (ii)
`lim_(x ->0) f(x) ≠ f(0)`
Function is discontinuous at x = 0
Hence we can remove the discontinuity by redefining f as
`f(x) = lim_(x ->0) (sin^2 5x)/x^2` for x ≠ 0
= 25 for x = 0
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