Let ' a ' be a positive constant number. Consider two curves...

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 $\lim _{a \rightarrow 0} \frac{S}{a^{2}}$
equals
(a) 4
(b) $1 / 2$
(c) 0
(d) $1 / 4$

Key Concepts

Area Between Curves

This concept involves finding the region enclosed by two or more curves by setting up and evaluating definite integrals. It is a fundamental application of integration in calculus, where the area is determined by integrating the difference between the functions over a given interval. The process often includes finding the points where the curves intersect to establish limits of integration.

Limit Evaluation in Calculus

Evaluating limits, particularly when a parameter approaches a specific value, is a key technique in calculus. This concept is crucial when determining the behavior of functions near certain points, which may include using L'Hôpital's rule, squeezing arguments, or algebraic simplification to resolve indeterminate forms.

Taylor Series and Asymptotic Approximations

When analyzing the behavior of functions for small variations in parameters, Taylor series expansions provide a powerful tool to approximate complex functions by polynomials. This technique is widely used to simplify expressions and to extract leading-order terms that dominate the behavior of the function in a limit, thus making it easier to compute limits and understand the function’s local behavior.

Properties of Exponential Functions

Exponential functions have unique properties such as constant proportional growth or decay, which make them integral to many problems in calculus. Their well-defined behavior under transformation and their simple derivative and integral forms facilitate a wide range of analyses, including solving differential equations, evaluating integrals, and performing approximations.

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