Let `f(x)={(1-cosx)/(x^2),\ \ \ w h e n\ x!=0 1,\ \ \ \ w h e n\ x=0` . Show that `f(x)` is discontinuous at `x=0` .

Explanation

`f(x)= {(1-cosx)/(x^2)` when `x!=0` and ` 1` when `x=0`

For the continuity at `x=0`

`f(0)=1`

`text { LHL }=lim _{x -> 0} f(x) `

`text{LHL}=lim _{h -> 0} f({0}-h) `

`text{LHL}=lim _{h -> 0} (1-cos(0-h))/((0-h)^2)`

`text{LHL} = lim _{h -> 0} (1-cos(-h))/(h^2)`

`text{LHL} = lim _{h -> 0} (1-cos(h))/(h^2)`

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