Animated Solution for Mathematics - Limits, Continuity and Differentiability: If $f(x)$ is differentiable and strictly increasing function, then the value of ${\lim }_{x\to 0}\frac{f(x^2)-f(x)}{f(x)-f(0)}$ is

Imagine you are standing on the edge of a mathematical landscape, looking at the function $f(x)$. We are told two vital things: it is differentiable, meaning it is smooth and continuous, and it is strictly increasing, meaning it is always climbing.

We are tasked with evaluating the limit:

At first glance, this looks like a daunting ratio, but let us break it down with the precision of a surgeon.

Before we panic, let us test the waters. If we substitute $x=0$ directly into our expression, we get:

This is the classic indeterminate form. It is not a dead end; it is an invitation to use L'Hospital's Rule. We are essentially looking for the rate at which the numerator vanishes compared to the denominator.

Since we have a $\frac{0}{0}$ form, we can differentiate the numerator and the denominator separately. This is where the Chain Rule becomes our most trusted ally.

For the numerator, we calculate:

For the denominator, we calculate:

Since $f(0)$ is a constant, its derivative is zero.

Now, we look at our new limit:

As $x$ approaches zero, the term $2x\cdot f^{\prime }(x^2)$ vanishes because $2(0)=0$. We are left with:

This is where the condition of being 'strictly increasing' saves us. Because $f(x)$ is strictly increasing, $f^{\prime }(0)$ must be greater than zero.

It is not zero, which means we can safely divide the numerator by the denominator. The $f^{\prime }(0)$ terms cancel out perfectly, leaving us with the elegant result of $-1$.