Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}\frac{x\cot (4x)}{{\sin }^{2}x{\cot }^2(2x)}$ is equal to:

The problem asks us to evaluate the limit:

This expression appears intimidating due to the presence of trigonometric functions in the denominator and numerator. However, we can simplify this by converting the cotangent functions into their tangent counterparts.

We utilize the fundamental identity $\cot \theta =\frac{1}{\tan \theta }$. Substituting this into our expression, we obtain:

By rearranging the terms to bring the denominator of the denominator to the numerator, the expression simplifies to:

To solve this, we rely on the standard limit theorems for small angles, specifically:

Our goal is to force the expression into these forms by introducing the necessary factors of $x$ in the numerator and denominator.

We manipulate the expression to align with the standard limits by multiplying and dividing by the appropriate constants:

To balance the angles, we rewrite the expression as:

As $x\to 0$, the terms $\frac{4x}{\tan (4x)}$, $\frac{\tan (2x)}{2x}$, and $\frac{\sin x}{x}$ all approach $1$. Substituting these values into our balanced equation:

The $x^2$ terms cancel out, leaving us with:

The final value of the limit is $1$.