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### Section 2.3 : One-Sided Limits

1. 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.
1. $$a = - 4$$
2. $$a = - 1$$
3. $$a = 2$$
4. $$a = 4$$
Solution
2. 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.
1. $$a = - 2$$
2. $$a = 1$$
3. $$a = 3$$
4. $$a = 5$$
Solution
3. Sketch a graph of a function that satisfies each of the following conditions. $\mathop {\lim }\limits_{x \to {2^{\, - }}} f\left( x \right) = 1\hspace{0.75in}\mathop {\lim }\limits_{x \to {2^{\, + }}} f\left( x \right) = - 4\hspace{0.75in}f\left( 2 \right) = 1$ Solution
4. 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