Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}\left(\left(\frac{1-{\cos }^2(3x)}{{\cos }^3(4x)}\right)\left(\frac{{\sin }^3(4x)}{({\log }_e(2x+1){)}^5}\right)\right)$ is equal to

Imagine you are standing before a complex limit expression, staring at a jumble of trigonometric and logarithmic functions. It looks like a chaotic mess, but in the world of JEE Advanced, chaos is just an opportunity for order.

We are evaluating the following limit:

The First Simplification

The first thing that should catch your eye is the term $1-{\cos }^2(3x)$. It is a classic trigonometric identity in disguise.

We know that ${\sin }^2(\theta )+{\cos }^2(\theta )=1$, which implies $1-{\cos }^2(\theta )={\sin }^2(\theta )$. By setting $\theta =3x$, our numerator instantly transforms into ${\sin }^2(3x)$.

The expression now looks much friendlier:

The Standard Limit Toolkit

Now, we need our secret weapons. We rely on two fundamental limits:

Notice that ${\cos }^3(4x)$ is harmless. As $x\to 0$, $\cos (4x)\to 1$, so ${\cos }^3(4x)\to 1$. It will not cause any division-by-zero issues.

The Balancing Act

This is the most crucial part of the journey. To use our standard limits, we must force the denominators to match the arguments of our functions.

For ${\sin }^2(3x)$, we need $(3x{)}^2=9x^2$ in the denominator. Thus, we write:

We apply the same logic to the other terms. For ${\sin }^3(4x)$, we need $(4x{)}^3=64x^3$ in the denominator. For $({\log }_e(1+2x){)}^5$, we need $(2x{)}^5=32x^5$ in the denominator.

The Grand Cancellation

Now, let us assemble our pieces back into the limit:

Look at the powers of $x$. In the numerator, we have $x^2\cdot x^3=x^5$. In the denominator, we have $x^5$. They cancel out perfectly.

The indeterminate form vanishes, leaving us with constants. As $x\to 0$, all the standard limit fractions evaluate to $1$.

We are left with:

And there it is—the elegance of mathematics in action. We started with a terrifying expression and ended with a simple, clean 18. Keep practicing this art of deconstruction, and no limit will ever scare you again.