How To Classify Discontinuities - Math Warehouse

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Using the tables below, what type of discontinuity seems to exist at $$x = 5$$?

$$ \begin{array}{c|lcc|l} {x} & {f(x)}\\ \hline 4.9 & 8.15\\ 4.99 & 8.015\\ 4.999 & 8.0015\\ 4.9999 & 8.00015\\ 4.99999 & 8.000015\\ \end{array} $$

$$ \begin{array}{c|lcc|l} {x} & {f(x)}\\ \hline 5.1 & 2.4\\ 5.01 & 2.43\\ 5.001 & 2.403\\ 5.0001& 2.4003\\ 5.00001 & 2.40003 \end{array} $$

Step 1

Examine the one-sided limits.

The table on the left tells us $$\lim\limits_{x\to5^-}f(x) \approx 8$$

The table on the right tells us $$\lim\limits_{x\to5^+}f(x) \approx 2.4$$

Answer

The tables lead us to believe the one-sided limits are different, so we conclude the function likely has a jump discontinuity at $$x = 5$$.