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State where the function is continuous, discontuous and the type of discontinuity for each

State where the function is continuous, discontuous and the type of discontinuity-example-1
User TommyVee
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f(x)=\begin{cases}(x)/(x-2);x<2 \\ 6;2\leq x\leq6 \\ (x-2)/(x^2-6x+8);x>6\end{cases}

For x = 2


\lim _(x->2^-)((x)/(x-2))=-\infty
\begin{gathered} \lim _(x->2^+)(6)=6 \\ \end{gathered}

Since:


\lim _(x->2^+)f(x)\\e\lim _(x->2^-)f(x)

The function is discontinuous at x = 2. Besides since the function has a vertical asymptote on one of the sides, we can conclude it is a infinite discontinuity.

For x = 6:


\lim _(x->6^-)6=6
\lim _(x->6^+)(x-2)/(x^2-6x+8)=\lim _(x->6^+)(x-2)/((x-2)(x-4))=\lim _(x->6^+)(1)/(x-4)=(1)/(2)

Since:


\lim _(x->6^+)f(x)\\e\lim _(x->6^-)f(x)

The function is discontinuous at x = 6, at this point the function has a jump discontinuity

User Wibble
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