# How do you show that the function h(x)=xe^sinx is continuous on its domain and what is the domain?

Nov 18, 2016

$h \left(x\right) = x {e}^{\sin} x$

$\sin x$ is continuous over $\mathbb{R}$ and it's domain is $x \in R$, and it's range is $x \in \left[- 1 , 1\right]$

${e}^{x}$ is continuous over $\mathbb{R}$ and it's domain is $x \in \mathbb{R}$, and it's range is {X in RR | x>0}.

$x$ is continuous over $\mathbb{R}$ and it's domain is $x \in \mathbb{R}$, and it's range is $x \in \mathbb{R}$.

Consequently, ${e}^{\sin} x$ is continuous over $\mathbb{R}$, and it's range is $x \in \mathbb{R}$, and it's domain is $\left\{x \in \mathbb{R} | {e}^{-} 1 \le x \le e\right\}$

Hence , $h \left(x\right) = x {e}^{\sin} x$ is continuous over $\mathbb{R}$, and it's range is $x \in \mathbb{R}$. and it's domain is $x \in \mathbb{R}$.

In fact $h \left(x\right)$ oscillates between $y = {e}^{-} 1$ and $y = e x$