Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}\frac{x(e^{\sqrt{1+x^2+x^4}-1}-1)}{\sqrt{1+x^2+x^4}-1}$

Welcome, future engineer. Today, we are going to dismantle a limit problem that looks like a monster but behaves like a kitten once you know its secret.

When you first look at the expression:

Your brain might instinctively panic. It looks messy, containing square roots, powers of four, and an exponential function all tangled together. But in the world of JEE Advanced, complexity is often just a mask for simplicity. The first step to mastery is not calculation; it is observation.

Notice the term $\sqrt{1+x^2+x^4}-1$. It appears in the exponent of $e$ and again in the denominator. This is not a coincidence; it is a breadcrumb trail left by the examiner.

Whenever you see a repeating, complex expression, do not fight it—embrace it. Let us define a new variable, $u$, such that:

Now, we must check the behavior of our new variable. As $x\to 0$, what happens to $u$? Substituting $x=0$ into our definition, we get $u=\sqrt{1+0+0}-1=1-1=0$.

This is crucial! Because $u\to 0$, we have unlocked the door to one of the most powerful tools in calculus: the standard limit.

With our substitution, the original limit transforms into something beautiful:

We know from our standard limit library that:

This is the moment where the 'scary' part of the problem simply vanishes. It collapses into unity.

Here is where many brilliant students stumble. They get so excited about solving the fraction that they forget the leading $x$.

We are left with ${\lim }_{x\to 0}x\cdot 1$. As $x$ approaches zero, the value of $x$ is zero. Therefore:

If that leading $x$ were not there, the answer would have been $1$. But it is there, and it changes everything.

This problem teaches us a vital lesson: never lose sight of the 'big picture' while you are busy solving the 'small details.' Stay focused, stay calm, and keep looking for those patterns. The final answer is 0.