Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{t\to 0}(1^{{\text{cosec}}^{2}t}+2^{{\text{cosec}}^{2}t}+\cdots +n^{{\text{cosec}}^{2}t}{)}^{{\sin }^{2}t}$ is equal to

A Lesson in Dominance

Imagine you are standing before a mathematical mountain. The expression ${\lim }_{t\to 0}(1^{{\text{cosec}}^{2}t}+2^{{\text{cosec}}^{2}t}+\cdots +n^{{\text{cosec}}^{2}t}{)}^{{\sin }^{2}t}$ looks intimidating, doesn't it?

It feels like a chaotic jumble of trigonometric functions and powers. But in the world of JEE Advanced, intimidation is just a mask for elegance. Let's peel back that mask together.

The Bridge of Substitution

The first thing that should catch your eye is the relationship between the exponents. We have ${\text{cosec}}^{2}t$ inside and ${\sin }^{2}t$ outside.

If you have been practicing your trigonometry, you know that ${\text{cosec}}^{2}t=\frac{1}{{\sin }^{2}t}$. This is our bridge.

Let us define a new variable, $y={\text{cosec}}^{2}t$. As $t\to 0$, ${\sin }^{2}t$ shrinks to zero. Consequently, $y=\frac{1}{{\sin }^{2}t}$ explodes toward infinity.

Our problem transforms from a tricky trigonometric limit into a beautiful algebraic limit:

The Law of the Jungle

Now, look at the expression $(1^y+2^y+\cdots +n^y{)}^{\frac{1}{y}}$. We have a sum of terms, all raised to the power $y$.

As $y$ grows toward infinity, which term do you think dictates the behavior of the sum? Think of it like a race. $n^y$ is the fastest runner, while $(n-1{)}^y$ cannot keep up.

As $y$ becomes massive, the gap between $n^y$ and $(n-1{)}^y$ becomes an abyss. In mathematics, we call this the Dominant Term.

To solve this, we must force the dominant term to reveal itself. We factor out $n^y$ from the parenthesis:

The Final Collapse

Now, watch the magic happen. We distribute the outer power $\frac{1}{y}$ to both factors.

The first factor, $n^y$, raised to the power $\frac{1}{y}$, simplifies beautifully to just $n$. The second factor is the bracketed sum raised to the power $\frac{1}{y}$.

Consider the terms inside the bracket. For any base $k<n$, the fraction $\frac{k}{n}$ is less than $1$.

When you raise a fraction less than $1$ to the power of infinity, it vanishes to zero. So, as $y\to \infty$, all those terms $(\frac{k}{n}{)}^y$ die out, leaving only the $1$.

We are left with $n\cdot (1{)}^{\frac{1}{y}}$. As $y\to \infty$, the exponent $\frac{1}{y}$ becomes $0$, and $1^0$ is simply $1$.

The final result is $n$.

This problem teaches us a vital lesson: never be afraid of complexity. When you see a sum of powers, look for the dominant term.

It is the key that unlocks the entire structure. You have just mastered a pattern that will serve you well in many future problems.

Keep this intuition sharp, and you will find that even the most intimidating limits are just puzzles waiting to be solved.