Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $k\in \mathbb{R}$. If ${\lim }_{x\to 0^{+}}(\sin (\sin kx)+\cos x+x{)}^{2/x}=e^6$, then the value of $k$ is

The given limit is:

At first glance, it looks like a complex arrangement of trigonometric functions and powers. However, in the context of JEE Advanced, such complexity is often a mask for underlying elegance.

Before diving into the algebra, we must identify the nature of the limit. As $x\to 0^{+}$, the base approaches $\sin (0)+\cos (0)+0=0+1+0=1$.

Simultaneously, the exponent $\frac{2}{x}$ approaches $\infty$. We are dealing with a classic $1^{\infty }$ indeterminate form, which requires a specific transformation.

To handle the variable in the exponent, we use the standard identity:

Applying this to our expression, we transform the limit into:

Since the result is $e^6$, we equate the exponents:

Dividing both sides by $2$, we obtain the simplified limit:

Checking the form again by substituting $x=0$, we obtain $\frac{0}{0}$. This confirms we should apply L'Hopital's Rule.

Differentiating the numerator and denominator with respect to $x$: 1. The derivative of the denominator $x$ is $1$. 2. Using the chain rule, the derivative of $\sin (\sin kx)$ is $\cos (\sin kx)\cdot \cos (kx)\cdot k$. 3. The derivative of $\cos x$ is $-\sin x$, and the derivative of $x$ is $1$.

This yields the following expression:

With the denominator now a constant, the indeterminate form has vanished. We substitute $x=0$ directly into the expression:

Since $\sin (0)=0$ and $\cos (0)=1$, the equation simplifies to:

This results in the simple linear equation:

Solving for $k$, we find the final answer: $k=2$