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If (lim)(x->c)(f(x)-f(c))/(x-c) exists f...

If `(lim)_(x->c)(f(x)-f(c))/(x-c)` exists finitely, write the value of `(lim)_(x->c)f(x)` .

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Verified by Experts

We know that `f'(x)=(lim)_(x->c)(f(x)-f(c))/(x-c)`
And we write,
`(lim)_(x->c)f(x)=(lim)_(x->c)[{(f(x)-f(c))/(x-c)}(x-c)+f(c)]`
`=(lim)_(x->c)[{(f(x)-f(c))/(x-c)}(x-c)]+f(c)`
`=(lim)_(x->c)(f(x)-f(c))/(x-c)(lim)_(x->c)(x-c)+f(c)`
`=f'(c)xx0+f(c)`
`=f(c)`
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