`lim_(x->0) (g(x)-g(c))/(x-c)`

`lim_(x->0) (sqrt(|x|))/x`

` ` In order to evaluate the above function, both the left hand limit and right hand limit need to be evaluated.

`L.H.L. = lim_(x->0^-) (sqrt(|x|))/(x) = lim_(x->0^-) (sqrt(-x))/(x) = DNE`

` `

`R.H.L. = lim_(x->0^+) (sqrt(|x|))/(x) = lim_(x->0^+) (sqrt(x))/(x) = DNE`

``Because the L.H.L. and...

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`lim_(x->0) (g(x)-g(c))/(x-c)`

`lim_(x->0) (sqrt(|x|))/x`

` ` In order to evaluate the above function, both the left hand limit and right hand limit need to be evaluated.

`L.H.L. = lim_(x->0^-) (sqrt(|x|))/(x) = lim_(x->0^-) (sqrt(-x))/(x) = DNE`

` `

`R.H.L. = lim_(x->0^+) (sqrt(|x|))/(x) = lim_(x->0^+) (sqrt(x))/(x) = DNE`

``Because the L.H.L. and R.H.L. both do not exist, the limit at x=0 does not exist, and therefore the derivative at x=0 does not exist.

` `

`g(x)=sqrt(abs x) `

`g'(x)=lim_(h->0) (f(x+h)-f(x))/h`

`g'(x)=lim_(h->0) ((x+h)^(1/2)-(x)^(1/2))/h`

at x=0

`g'(x)=lim_(h->0) h^(1/2)/h`

`g'(x)=lim_(h->0) 1/h^(1/2)`

So, the limit is undefined , g'(x) does not exists.