Animated Solution for Mathematics - Limits, Continuity and Differentiability: If ${\lim }_{x\to \infty }{\left(1+\frac{a}{x}+\frac{b}{x^2}\right)}^{2x}=e^2$, then the values of $a$ and $b$, are

Welcome back, future IITians! Today, we are going to dissect a problem that often trips up students in the JEE Advanced arena. It is not because the math is impossible, but because it tests your ability to see through the noise.

We are looking at the limit:

At first glance, this looks like a standard exponential limit, but let's peel back the layers.

First, we must understand the behavior of the base as $x$ grows without bound. As $x\to \infty$, the terms $\frac{a}{x}$ and $\frac{b}{x^2}$ both shrink to zero. This leaves us with a base approaching $1$.

Simultaneously, the exponent $2x$ is racing toward infinity. This is the classic $1^{\infty }$ indeterminate form.

Whenever you see this, do not panic. It is a signal to use the standard limit property:

This formula is your best friend in calculus. It transforms a terrifying exponential limit into a simple product in the exponent.

Let us apply this property to our problem. Here, $f(x)=1+\frac{a}{x}+\frac{b}{x^2}$ and $g(x)=2x$.

Substituting these into our formula, we get:

Notice the elegance here! The $1$ and $-1$ inside the bracket cancel out perfectly, leaving us with:

Now, we distribute the $2x$ into the parentheses. This is where the magic happens:

Our exponent is now ${\lim }_{x\to \infty }(2a+\frac{2b}{x})$.

As $x\to \infty$, the term $\frac{2b}{x}$ approaches zero, regardless of the value of $b$. This is the crucial realization: $b$ has no impact on the limit!

The limit of the exponent is simply $2a$. Thus, our entire expression simplifies to $e^{2a}$.

We are given that this limit equals $e^2$. By equating the exponents, $2a=2$, we find $a=1$.

Since $b$ disappeared during the limit process, it can be any real number. Therefore, the final solution is $a=1$ and $b\in \mathbb{R}$.

You have just conquered a classic JEE limit problem. Keep practicing, stay curious, and keep pushing your boundaries!