Question

Sketch the graph of any function \(f\) such that \(\lim _{x \rightarrow 3^{+}} f(x)=1\) and \(\quad \lim _{x \rightarrow 3^{-}} f(x)=0\). Is the function continuous at \(x=3\) ? Explain.

Short answer

Based on the graph and the concept of limits, the function \(f(x)\) is not continuous at \(x=3\). The reason is that the left-hand and right-hand limits at \(x=3\) are not equal.

Step-by-step solution

Identify the Limit Conditions

From the given conditions, the function's behavior around \(x=3\) is known. When \(x\) approaches \(3\) from the right (positive side), the function approaches a value of \(1\). However, when \(x\) approaches \(3\) from the left (negative side), the function tends towards \(0\). These are the two points that will be plotted on the graph.

Sketch the Graph

Draw a graph with an x-axis and y-axis. Mark the point (3,0) on the graph, this is the left-hand limit point. Similarly, mark the point (3,1), this is the right-hand limit point. Draw a filled circle at (3,0) and an open circle at (3,1). Draw line segments approaching these points from their respective sides (from left for (3,0), and from right for (3,1)).

Analyze for Continuity

A function is considered continuous at a certain point if the left-hand and right-hand limits exist and are equal to the function's value at that point. In this case, the left-hand limit (0) and the right-hand limit (1) are not the same, indicating that the function is not continuous at \(x=3\).