Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\), \(\mathop {\lim }\limits_{x \to {a^{\, - }}} f\left( x \right)\), \(\mathop {\lim }\limits_{x \to {a^{\, + }}} f\left( x \right)\), and \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\). If any of the quantities do not exist clearly explain why.
Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\), \(\mathop {\lim }\limits_{x \to {a^{\, - }}} f\left( x \right)\), \(\mathop {\lim }\limits_{x \to {a^{\, + }}} f\left( x \right)\), and \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\). If any of the quantities do not exist clearly explain why.
Sketch a graph of a function that satisfies each of the following conditions.
\[\mathop {\lim }\limits_{x \to {2^{\, - }}} f\left( x \right) = 1\hspace{0.75in}\mathop {\lim }\limits_{x \to {2^{\, + }}} f\left( x \right) = - 4\hspace{0.75in}f\left( 2 \right) = 1\]
Solution
Sketch a graph of a function that satisfies each of the following conditions.
\[\begin{array}{ccl}\mathop {\lim }\limits_{x \to {3^{\, - }}} f\left( x \right) = 0 & \hspace{0.5in}\mathop {\lim }\limits_{x \to {3^{\, + }}} f\left( x \right) = 4 & \hspace{0.5in}f\left( 3 \right){\mbox{ does not exist}}\\ \mathop {\lim }\limits_{x \to - 1} f\left( x \right) = - 3 & \hspace{0.5in} f\left( { - 1} \right) = 2 & \end{array}\]
Solution
