Question

Suppose that f'(4) = 3 , g'(4) = 7 , g(4) = 4 and g(x) not= to 4 for xnot= to 4 . then cmpute lim xgose to 4 f(g(x))/x-4-f(4)/x-4

Answer 1

It looks like the limit you want to compute is

`\displaystyle \lim_(x\to4) (f(g(x)) - f(4))/(x-4)`

Since
`g(4)=4`, this limit corresponds exactly to the derivative of
`(f\circ g)(x) = f(g(x))` at
`x=4`. Recall that

`f'(a) = \displaystyle \lim_(x\to a) (f(x) - f(a))/(x - a)`

By the chain rule,

`(f\circ g)'(4) = f'(g(4)) * g'(4) = f'(4) * g'(4) = 3*7 = \boxed{21}`

Since
`g'(4)` exists,
`g` is differentiable at
`x=4` so it must be continuous.