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Is the following function continuous? $$f(x)=\left\{\begin{array}{c}x+3 \text { for } x<2 \\ 5 \text { for } x=2 \\ x^{4}-11 \text { for } x>2\end{array}\right.$$
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A Since this function is a junction of two continuous functions, we only have to worry about discontinuity at the point where the functions meet, i.e. at $x=2$.

$\lim _{x \rightarrow 2^{2}} f(x)=\lim _{x \rightarrow 2-} x+3=2+3=5$

$\lim _{x \rightarrow 2^{2+}} f(x)=\lim _{x \rightarrow 2^{+}}\left(x^{4}-11\right)=2^{4}-11=5$

Thus, $\lim _{x \rightarrow 2} f(x)=5$, and $f(2)=5$, so $\lim _{x \rightarrow 2} f(x)=f(2)$. It follows that $f$ is continuous.
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