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Section 2.3 : One-Sided Limits

3. Sketch a graph of a function that satisfies each of the following conditions.

\[\mathop {\lim }\limits_{x \to {2^{\, - }}} f\left( x \right) = 1\hspace{0.75in}\mathop {\lim }\limits_{x \to {2^{\, + }}} f\left( x \right) = - 4\hspace{0.75in}f\left( 2 \right) = 1\] Show Solution

There are literally an infinite number of possible graphs that we could give here for an answer. However, all of them must have a closed dot on the graph at the point \(\left( {2,1} \right)\), the graph must be approaching a value of 1 as it approaches \(x = 2\) from the left (as indicated by the left-hand limit) and it must be approaching a value of -4 as it approaches \(x = 2\) from the right (as indicated by the right-hand limit).

Here is a sketch of one possible graph that meets these conditions.