The value of limx_→∞xⁿ+nxⁿ⁻¹+1/e^[x], n∈1 is (where [.] denotes greatest integer function)

Solution for Limit Problem

lim_x→∞ (xⁿ + n x^n-1 + 1/e [x]) where n ∈ 1 (and [.] denotes the greatest integer function).

Step-by-Step Solution:

1. Breaking the Expression into Parts:

The given expression has three parts:

• xⁿ
• n x^n-1
• 1/e [x]

We need to evaluate the behavior of each of these terms as x approaches infinity.

2. Term 1: xⁿ

As x → ∞, the term xⁿ grows very large because n is a positive integer. Specifically, it tends to infinity.

3. Term 2: n x^n-1

Similarly, n x^n-1 also grows very large, but at a slower rate compared to xⁿ. However, this term still tends to infinity as x → ∞.

4. Term 3: 1/e [x]

The greatest integer function [x] rounds x down to the nearest integer. As x → ∞, [x] will be very close to x, but always less than or equal to x. So, [x] behaves approximately like x as x → ∞, making the term 1/e [x] grow at the same rate as x/e, which increases indefinitely as x → ∞.

5. Combining the Terms:

As x → ∞, all the terms involved grow without bound, but the greatest integer function introduces a subtle change in the final value of the limit. Specifically, 1/e [x] will slightly modify the sum, but the primary growth term is still dominated by xⁿ.

6. Impact of the Greatest Integer Function:

The greatest integer function [x] only affects the fractional part of x. As x → ∞, the fractional part of x becomes insignificant compared to the terms like xⁿ and n x^n-1. Thus, the limit behavior is determined primarily by the dominant powers of x.

7. Conclusion:

Since the highest power of x in the expression is xⁿ, and the greatest integer function affects the lower-order terms in a negligible way as x → ∞, the value of the limit approaches n.

Final Answer:

The value of the limit is n, and the correct option is:

b) zero is not correct because the behavior of the terms causes the value of the expression to grow towards n.

Thus, the value of the limit is n and the correct answer is c) n.