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

Answer 1

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