This applet contains three numerical examples of limits as [math]x \rightarrow 0[/math]. Click on a checkbox to see the graph of a particular example.[br][br]Example 1: We can see from the table (and the graph) that [math]\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{x+1}-1} = 2[/math] even though the function is not defined when [math]x = 0[/math].[br][br]Example 2: We can see from the table (and the graph) that [math]\displaystyle\lim_{x \rightarrow 0} \left\{ \begin{array}{ l l } 1, & x\neq0 \\ 2, & x = 0 \end{array} \right. = 1[/math], which disagrees with [math]f\left(0\right) = 2[/math].[br][br]Example 3: We can see from the table (and the graph) that [math]\displaystyle\lim_{x \rightarrow 0} \frac{|x|}{x}[/math] does not exist. The function is also not defined when [math]x = 0[/math].