Let $f$ be a continuous function taking positive real values, and set $G(x) =\int_x^{x^2} f(z)\,dz$. Find $G'(x)$. Estimate the value of $G(x)$ for values of $x$ near 1. (You should do more than just give $\lim_{x\to1}G(x)$.)
For $G'(x)$ I found $2x\lim_{z\to x^2}f(z))-(\lim_{z\to x}f(z)$ using the chain rule. I don't really get what I should be doing for $G(x)$ as $x\to1$, of course I should work on the different cases possibles: when $x$ tends to $1^-,1^+$, when $f$ is increasing, decreasing. But how can I compute the actual limit as I don't have $f(z)$. (Even though, intuitively I feel like it will be equal to 0 as $x^2 = x$ as $x\to1$.)
Using Leibniz's Rule, we get $$G'(x)=f(x^2)\frac{d}{dx}(x^2)-f(x)\frac{d}{dx}(x)=2xf(x^2)-f(x)$$
Using linear approximation (for $x$ in neighborhood of $1$):
$$G(x) \approx G(1)+G'(1)(x-1).$$
Observe that $G(1)=0$ and from above expression we obtain $G'(1)=2f(1)-f(1)=f(1)$. So, $$G(x) \approx f(1)(x-1).$$