Let 'a' be a positive constant number. Consider two curves `C_1: y=e^x, C_2:y=e^(a-x)`. Let S be the area of the part surrounding by `C_1, C_2` and the y axis, then `Lim_(a->0) s/a^2` equals (A) 4 (B) `1/2` (C) 0 (D) `1/4`

Options

A. 4
B. `1/2`
C. 0
D. `1.4`

Answer

To solve the problem, we need to find the area \( S \) bounded by the curves \( C_1: y = e^x \) and \( C_2: y = e^{a-x} \), and then evaluate the limit \( \lim_{a \to 0} \frac{S}{a^2} \).

### Step-by-Step Solution:

1. **Identify the Intersection Points**:
To find the area between the curves, we first need to determine their points of intersection. Set \( e^x = e^{a-x} \).
\[ e^x = e^{a-x} \implies x = a - x \implies 2x = a \implies x = \frac{a}{2} \]
The curves intersect at \( x = \frac{a}{2} \).

2. **Determine the Area \( S \)**:
The area \( S \) between the curves from \( x = 0 \) to \( x = \frac{a}{2} \) can be expressed as:
\[ S = \int_0^{\frac{a}{2}} (e^{a-x} - e^x) \, dx \]

3. **Evaluate the Integral**:
We can split the integral:
\[ S = \int_0^{\frac{a}{2}} e^{a-x} \, dx - \int_0^{\frac{a}{2}} e^x \, dx \]
- For the first integral:
\[ \int e^{a-x} \, dx = -e^{a-x} \Big|_0^{\frac{a}{2}} = -e^{a-\frac{a}{2}} + e^a = -e^{\frac{a}{2}} + e^a \]
- For the second integral:
\[ \int e^x \, dx = e^x \Big|_0^{\frac{a}{2}} = e^{\frac{a}{2}} - 1 \]
Thus,
\[ S = \left(-e^{\frac{a}{2}} + e^a\right) - \left(e^{\frac{a}{2}} - 1\right) = e^a - 2e^{\frac{a}{2}} + 1 \]

4. **Simplify the Expression for \( S \)**:
We can rewrite \( S \):
\[ S = e^a - 2e^{\frac{a}{2}} + 1 \]

5. **Evaluate the Limit**:
Now we need to find \( \lim_{a \to 0} \frac{S}{a^2} \):
\[ S = e^a - 2e^{\frac{a}{2}} + 1 \]
Using the Taylor series expansion:
- \( e^a \approx 1 + a + \frac{a^2}{2} \)
- \( e^{\frac{a}{2}} \approx 1 + \frac{a}{2} + \frac{a^2}{8} \)

Substituting these approximations:
\[ S \approx \left(1 + a + \frac{a^2}{2}\right) - 2\left(1 + \frac{a}{2} + \frac{a^2}{8}\right) + 1 \]
Simplifying:
\[ S \approx 1 + a + \frac{a^2}{2} - 2 - a - \frac{a^2}{4} + 1 = \frac{a^2}{4} \]

6. **Final Limit Calculation**:
Now, we evaluate:
\[ \lim_{a \to 0} \frac{S}{a^2} = \lim_{a \to 0} \frac{\frac{a^2}{4}}{a^2} = \frac{1}{4} \]

Thus, the final answer is:
\[ \boxed{\frac{1}{4}} \]