# Chapter 2 - Section 2.5 - Continuity - 2.5 Exercises: 28

$R(t)$ is continous on $R$.

#### Work Step by Step

$$R(t)=\frac{e^{\sin t}}{2+\cos\pi t}$$ 1) Consider the numerator We see that $\sin t$ is defined for $\forall t\in R$. So, $e^{sin t}$ is also defined for $\forall t\in R$. Therefore, the domain of $e^{\sin t}$ is $R$. 2) Consider the denominator We know $2+ \cos\pi t$ is defined for $\forall t\in R$. Therefore, the domain of $2+\cos\pi t$ is $R$. 3) Find the common domain of both the numerator and denominator. Combine the results from 1) and 2), we conclude the common domain is $R$. In other words, according to Theorem 7, both $e^{\sin t}$ and $2+\cos\pi t$ are continuous on $R$. 4) According to Theorem 4, $R(t)$ is also continous on $R$ except where $2+\cos\pi t=0$. However, notice that $$-1\leq\cos\pi t\leq1$$$$1\leq2+\cos\pi t\leq3$$ for $\forall t\in R$ Which means, $2+\cos\pi t\ne0$ for $\forall t\in R$. In conclusion, $R(t)$ is continous on $R$.

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