Question

Let $y=f(x)$ satisfies the differential equation $(\sin x) d y+(\cos x) y d x=e^x d x$ with $f(0)=0$. Then the value of $\displaystyle\lim _{x \rightarrow 0} f(x)$ is

• A. $0$
• B. $1$
• C. $-1$
• D. Does not exist

Answer 1

Answer: (B)

$(\sin x) d y+(\cos x) y d x=e^x d x$
$d[(\sin x) \cdot y]=e^x d x$
Integrating $y \sin x=e^x+c$
Putting $f(0)=0$
$0=1+c \Rightarrow c=-1$
$y=\frac{e^x-1}{\sin x}$
$\displaystyle\lim _{x \rightarrow 0} \frac{e^x-1}{\sin x}=\displaystyle\lim _{x \rightarrow 0} \frac{e^x-1}{x} \frac{x}{\sin x}=1$