Find the points of discontinuity, if any, of the following function: `f(x)={x^3-x^2+2x-2\ \ \ ,\ \ \ if\ x!=1 4\ \ \ ,\ \ \ if\ x=1`

Explanation

Given function `f(x)= (x^{3}-x^{2}+2 x-2), & text { if } x neq 1 4, & text { if } x=1`


Left hand limit at `x=1: lim _{x -> 1^{-}} f(x)`


`quad=lim _{{x} -> 1^{-}}({x}^{3}-{x}^{2}+2 {x}-2)`


`=lim _{{h} -> 0}{(1-{h})^{3}-(1-{h})^{2}+2(1-{h})-2}`


`=lim _{{h} -> 0}(1-{h})^{3}-lim _{{h} -> 0}(1-{h})^{2}+2 lim _{{h} -> 0}(1-{h})-2`


`=1-1+2-2` `=0`


Right hand limit at `x=1: lim _{{x} -> 1^{+}} {f}({x})`


`=lim _{{h} -> 0}{(1+{h})^{3}-(1+{x}-{x}^{2}+2 {x}-2))^{2}+2(1+{h})-2}`


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