Question

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 -

• A. $\underset{a\to -\infty }{\mathrm{Lim}}S=1$
• B. $\underset{a\to 0}{\mathrm{Lim}}\frac{S}{a^2}=\frac{1}{4}$
• C. Range of $S$ is $[0,\infty )$
• D. S(a) is neither odd nor even

Answer 1

Answer: (A), (B), (C), (D)

$S=\underset{0}{\overset{a/2}{\int }}\left(e^{a-x}-e^x\right)dx$
$=-\left[2e^{a/2}-\left(e^a+1\right)\right]$
Now $\underset{a\to 0}{\lim }\frac{e^a-2e^{a/2}+1}{a^2}=\underset{a\to 0}{\lim }{\left(\frac{e^{a/2}-1}{a/2}\right)}^2\frac{1}{4}=\frac{1}{4}$