Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f:\mathbb{R}\to \mathbb{R}$ be a function. We say that $f$ has PROPERTY 1 if ${\lim }_{h\to 0}\frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite, and PROPERTY 2 if ${\lim }_{h\to 0}\frac{f(h)-f(0)}{h^2}$ exists and is finite. Then which of the following options is/are correct ?

In calculus, we often evaluate the behavior of a function $f(x)$ near the origin by comparing its growth rate to specific "yardsticks." We are investigating two properties based on the limit of the difference quotient:

Property 1: The limit $L_1={\lim }_{h\to 0}\frac{f(h)-f(0)}{\sqrt{|h|}}$ is finite.

Property 2: The limit $L_2={\lim }_{h\to 0}\frac{f(h)-f(0)}{h^2}$ is finite.

These limits determine if the function $f(h)-f(0)$ vanishes at least as fast as the denominator. If the numerator shrinks faster than the denominator, the limit is zero; if they shrink at the same rate, the limit is a non-zero constant. If the denominator shrinks faster, the limit diverges to infinity.

Let us test $f(x)=x^{2/3}$ against Property 1. We set up the limit as follows:

Using the laws of exponents, we subtract the powers:

Thus, the limit becomes ${\lim }_{h\to 0}|h{|}^{1/6}=0$. Since zero is a finite number, $f(x)=x^{2/3}$ satisfies Property 1.

Next, consider $f(x)=|x|$ for Property 1. The limit is:

As $h$ approaches zero, this limit evaluates to $0$. Because the result is finite, $f(x)=|x|$ also satisfies Property 1.

Now, consider $f(x)=x|x|$ for Property 2. The limit is:

This limit is a classic case where the Left-Hand Limit ($-1$) and the Right-Hand Limit ($1$) disagree. Therefore, the limit does not exist, and $f(x)=x|x|$ fails Property 2.

Finally, for $f(x)=\sin x$ and Property 2, we evaluate:

We know that ${\lim }_{h\to 0}\frac{\sin h}{h}=1$, but ${\lim }_{h\to 0}\frac{1}{h}$ is undefined (tending to infinity). Consequently, the product is infinite, and $f(x)=\sin x$ fails Property 2.

We have observed how different functions behave under the microscope of these limits. We found that $f(x)=x^{2/3}$ and $f(x)=|x|$ satisfy Property 1, while $f(x)=x|x|$ and $f(x)=\sin x$ fail Property 2. Remember that these limits are simply comparisons of growth rates, a fundamental intuition for mastering calculus.