Calculus I - One-Sided Limits (Practice Problems)

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Section 2.3 : One-Sided Limits

Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\), \(\mathop {\lim }\limits_{x \to {a^{,円 - }}} f\left( x \right)\), \(\mathop {\lim }\limits_{x \to {a^{,円 + }}} f\left( x \right)\), 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
[画像:OneSidedLimit_Ex1]
Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\), \(\mathop {\lim }\limits_{x \to {a^{,円 - }}} f\left( x \right)\), \(\mathop {\lim }\limits_{x \to {a^{,円 + }}} f\left( x \right)\), 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
[画像:OneSidedLimit_Ex2]
Sketch a graph of a function that satisfies each of the following conditions. \[\mathop {\lim }\limits_{x \to {2^{,円 - }}} f\left( x \right) = 1\hspace{0.75in}\mathop {\lim }\limits_{x \to {2^{,円 + }}} f\left( x \right) = - 4\hspace{0.75in}f\left( 2 \right) = 1\] Solution
Sketch a graph of a function that satisfies each of the following conditions. \[\begin{array}{ccl}\mathop {\lim }\limits_{x \to {3^{,円 - }}} f\left( x \right) = 0 & \hspace{0.5in}\mathop {\lim }\limits_{x \to {3^{,円 + }}} f\left( x \right) = 4 & \hspace{0.5in}f\left( 3 \right){\mbox{ does not exist}}\\ \mathop {\lim }\limits_{x \to - 1} f\left( x \right) = - 3 & \hspace{0.5in} f\left( { - 1} \right) = 2 & \end{array}\] Solution

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