Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to \frac{\pi }{2}}\frac{1-\tan (x/2)}{1+\tan (x/2)}\frac{[1-\sin x]}{[\pi -2x{]}^3}$ is

Welcome, fellow traveler of the mathematical landscape! Today, we are standing before a limit that looks like a daunting fortress:

When you first encounter this, your instinct might be to plug in $x=\frac{\pi }{2}$ immediately. Let's see what happens.

The tangent term becomes $\frac{1-\tan (\pi /4)}{1+\tan (\pi /4)}=\frac{1-1}{1+1}=0$. The sine term becomes $1-\sin (\pi /2)=1-1=0$. The denominator becomes $(\pi -2(\pi /2){)}^3=0^3=0$.

We are staring at a classic $\frac{0}{0}$ indeterminate form. It is a mirage! It tells us nothing about the actual value of the limit. We need a better plan.

To dismantle this fortress, we need the right tools. Look closely at the first part of the expression: $\frac{1-\tan (x/2)}{1+\tan (x/2)}$.

This is not just a random collection of terms; it is a hidden identity. Recall the formula:

By setting $\theta =x/2$, this entire fraction collapses into a single, elegant term: $\tan (\frac{\pi }{4}-\frac{x}{2})$. Suddenly, the expression looks much less intimidating.

Now, let's address the limit itself. Approaching $\frac{\pi }{2}$ is awkward. What if we could shift our perspective?

Let's define a new variable, $y$, such that $x=\frac{\pi }{2}+y$. As $x\to \frac{\pi }{2}$, $y\to 0$. This is the secret weapon of limit problems!

Let's transform our terms:

1. Tangent term: $\tan (\frac{\pi }{4}-\frac{1}{2}(\frac{\pi }{2}+y))=\tan (\frac{\pi }{4}-\frac{\pi }{4}-\frac{y}{2})=\tan (-\frac{y}{2})=-\tan (\frac{y}{2})$. 2. Sine term: $1-\sin (\frac{\pi }{2}+y)=1-\cos y$. 3. Denominator: $(\pi -2(\frac{\pi }{2}+y){)}^3=(\pi -\pi -2y{)}^3=(-2y{)}^3=-8y^3$.

Now, let's reassemble our expression:

The negative signs cancel out, leaving us with:

We know that $1-\cos y=2{\sin }^2(y/2)$. Substituting this in, we get:

Now, we split the $y^3$ to match the angles:

To use the standard limit ${\lim }_{\theta \to 0}\frac{\tan \theta }{\theta }=1$, we need the denominator to be $y/2$. We adjust:

This simplifies to:

We have arrived at the destination! The limit is $\frac{1}{32}$. Remember, every complex limit is just a series of small, logical steps waiting to be taken.