`h(x) = |x + 7|, c = -7` Use the alternate form of the derivative to find the derivative at x = c (if it exists)

Expert Answers

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`lim_(x->-7) (h(x) - h(c))/(x-c)`

` ` `lim_(x->-7) (|x+7|)/(x+7)`

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->-7^-) (|x+7|)/(x+7) = lim_(x->-7^-) (-(x+7))/(x+7) = -1`

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`R.H.L. = lim_(x->-7^+) (|x+7|)/(x+7) = lim_(x->-7^+) (x+7)/(x+7) = 1`

``Because the L.H.L. `!=` R.H.L., the limit at x=-7 does not exist, and therefore the derivative at x=-7 does not exist.

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`f(x)=|x+7|`

`f'(-7)=lim_(h->0)(f(-7+h)-f(-7))/h`

`f'(-7)=lim_(h->0) ((-7+h+7)-(-7+7))/h`

`f'(-7)=lim_(h->0) h/h`

Since the limit is undefined , f'(-7) is not defined.

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