Question

Please help me to solve this problem? It's about real analysis. I am studying it for preparing my midterm.. Thank you very much for your help.

Answer 1

Consider the function:

`h(x) = (f(x))/(g(x))`

we know that `h(x)` is continuous for `x in(a,b)` and `x !=c` because both `f(x)` and `g(x)` are continuous and `g(x) !=0`

Besides we know that:

`lim_(x-> c) h(x) = L in RR`

exists and is finite.

Also:

`lim_(x->c) g(x) = B in RR`

exists and is finite, but then:

`lim_(x->c) h(x) g(x) = L*B`

and since `B=0`:

`lim_(x->c) h(x) g(x) = 0`

However:

`h(x) g(x) = (f(x))/(g(x)) g(x) = f(x)`

so:

`lim_(x->c) f(x) = 0`

And as by hypothesis:

`lim_(x->c) f(x) = A`

then `A=0`.