Animated Solution for Mathematics - Limits, Continuity and Differentiability: The value of ${\lim }_{x\to 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to:

Facing the Eighth Root Monster

Welcome, future engineer. Today we are going to dismantle a problem that, at first glance, looks like a mathematical nightmare. We are looking at the limit:

Eighth roots? It sounds terrifying, doesn't it? But here is the secret: in the world of JEE Advanced, intimidation is just a test of your conceptual clarity. Let's break this down together.

The Diagnostic

Before we panic, let's perform a simple diagnostic. What happens when $x$ approaches $0$?

The numerator becomes $0$. The denominator becomes $\sqrt[8]{1-\sin (0)}-\sqrt[8]{1+\sin (0)}$, which is $\sqrt[8]{1}-\sqrt[8]{1}=0$.

We have a $0/0$ indeterminate form. This is our green light; it tells us that there is a hidden structure waiting to be revealed. We don't need to run away; we just need to simplify.

The Weaponry

How do we handle these eighth roots? We use the Binomial Approximation. Recall that for a small quantity $t$, $(1+t{)}^n\approx 1+nt$.

This is the scalpel that will cut through the complexity. Since $x\to 0$, $\sin x$ is also approaching $0$. This makes $\sin x$ our tiny quantity $t$.

We can rewrite our denominator using fractional exponents:

Now, we apply the approximation to each term. For the first term, $t=-\sin x$ and $n=1/8$:

For the second term, $t=\sin x$ and $n=1/8$:

The Transformation

Now, watch the magic happen. We substitute these approximations back into our denominator:

Let's simplify this carefully. The $1$ and $-1$ cancel out perfectly, leaving us with:

The complex roots have vanished, replaced by a simple trigonometric term.

The Final Act

Our limit now looks like this:

We can pull the constant $-4$ out of the limit:

We know the standard limit ${\lim }_{x\to 0}\frac{\sin x}{x}=1$, which means its reciprocal ${\lim }_{x\to 0}\frac{x}{\sin x}$ is also $1$. Therefore, our final answer is:

See? The monster was just a paper tiger. By staying calm and applying the right tools, you turned a terrifying expression into a simple, elegant result. Keep this mindset for every problem you face, and you will conquer the JEE.