Prove g'(x)=g(x): Hints & Solutions

[drawar]

Homework Statement

Let f and g be two functions on ℝ such that:
1. $g(x) = xf(x) + 1$ for all x,
2. $g(x + y) = g(x)g(y)$ for all x,y,
3. $\mathop {\lim }\limits_{x \to 0} f(x) = 1$.
Prove that: $g'(x) = g(x)$ for all x.

Homework Equations

The Attempt at a Solution

Sorry for asking this question without showing my workings but I just don't know how to get started. Any hints would be much appreciated. TIA!

 

[SammyS]

Homework Statement

Let f and g be two functions on ℝ such that:
1. $g(x) = xf(x) + 1$ for all x,
2. $g(x + y) = g(x)g(y)$ for all x,y,
3. $\mathop {\lim }\limits_{x \to 0} f(x) = 1$.
Prove that: $g'(x) = g(x)$ for all x.

Homework Equations

The Attempt at a Solution

Sorry for asking this question without showing my workings but I just don't know how to get started. Any hints would be much appreciated. TIA!

Just use the "limit" definition of the derivative.

 

[drawar]

Just use the "limit" definition of the derivative.

So I have to prove $\mathop {\lim }\limits_{h \to 0} \frac{{g(x + h) - g(x)}}{h} = g(x)$?

 

[SammyS]

So I have to prove $\mathop {\lim }\limits_{h \to 0} \frac{{g(x + h) - g(x)}}{h} = g(x)$?

Yes.

It works out quite nicely.

 

[happysauce]

Yes.

It works out quite nicely.

Yup, remember the 3 properties while you use the precise definition of derivative.

 

[drawar]

Thank you all, please check my working below:

$g'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{g(x + h) - g(x)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{g(x).g(h) - g(x)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{g(x).((g(h) - 1)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{g(x)hf(h)}}{h} = \mathop {\lim }\limits_{h \to 0} g(x)f(h) = \mathop {\lim }\limits_{h \to 0} g(x).\mathop {\lim }\limits_{h \to 0} f(h) = \mathop {\lim }\limits_{h \to 0} g(x) = g(x)$

 

[Crosstalk]

That looks correct to me.