NIMCET 2008 — Mathematics PYQ

Step 1: Write the expansions

• $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$

• $e^x=1+x+\frac{x^2}{2!}+\dots$

Step 2: Simplify the terms in the numerator

First term:

Second term:

The leading term here is $-x$.

Step 3: Combine the terms in the limit

Now, multiply the leading terms of both parts:

The limit expression becomes:

Step 4: Determine the value of $n$

For the limit to be a finite non-zero number, the power of $x$ in the denominator must match the lowest power of $x$ in the numerator.

• If n < 3, the limit is $0$.

• If n > 3, the limit is infinite.

• If $n=3$, the limit is:

  $$\underset{x\to 0}{\lim }\frac{x^3/2}{x^3}=\frac{1}{2}$$

  Since $\frac{1}{2}$ is a finite non-zero number, $n$ must be 3.

Final Answer:

The value of $n$ is $3$.

The correct option is (c).