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# Decide whether the function is continuous on the given interval. If not, define or redefine the function at one point to make it continuous everywhere on the given interval, or else explain why that cannot be done.$f(x)=\frac{x+4}{x-8} \text { on }[0,10]$

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The rational function $f ( x ) = \frac { x + 4 } { x - 8 }$ is conntinuous everywhen $x-8\neq 0$. In other words, $f(x)$ is continuous everywhere except at $x=8$. The limit at this point does not exist because

\begin{aligned} \lim _ { x \rightarrow 8 ^ { - } } f ( x ) = \lim _ { x \rightarrow 8 ^ { - } } \frac { x + 4 } { x - 8 }=-\infty\ \ \ x<8\Rightarrow x-8<0\\ \Rightarrow \ \text{the argument decreases without bound when x approaches 8 from the left} \end{aligned}

and

\begin{aligned} \lim _ { x \rightarrow 8 ^ { + } } f ( x ) = \lim _ { x \rightarrow 8 ^+} \frac { x + 4 } { x - 8 } = +\infty \ \ \ \text{because}\ x>8\ \text{ and the argument increases without bound} \end{aligned}

Since the limit at $x = 8 \in [ 0,10 ]$ does not exist, $f(x)$ is discontinuous on the prescribed interval

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