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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. \(a = - 5\)
    2. \(a = - 2\)
    3. \(a = 1\)
    4. \(a = 4\)
    This graph consists of two V shaped portions.  The first portion is in the domain \(-7 \le x < 1\).  Its point is at (-5,7) – this is an open dot, and opens downwards.  There is a closed dot at (-5,3) and at (-2,2).  The graph ends on the right at an open dot at (1,-3).  The second portion is in the domain \(1 < x \le 6\).  Its point is at (4,-2) – this is an open dot, and opens upwards.  The graph starts at an open dot at (1,4) and there is a closed dot at (1,6).
  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. \(a = - 1\)
    2. \(a = 1\)
    3. \(a = 3\)
    This graph consists of four horizontal line segments.  The first is in the domain \(x < -1\) and is at y=4.  It ends at a closed dot of (-1,4).  The second is in the domain \(-1 < x < \le 1\) and is at y=-2.  It starts at an open dot at (-1,-2) and ends at a closed dot at (1,-2).  The third is in the domain \(1 < x < 3\) and is at y=3.  It starts at an open dot of (1,3) and ends at an open dot of (3,3).  The final segment is in the domain \(x>3\) and is at y=1.  It starts with a closed dot at (3,1).
  3. 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. \(a = - 3\)
    2. \(a = - 1\)
    3. \(a = 1\)
    4. \(a = 2\)
    This graph consists of four segments.  The first is in the domain \(x < -3\).  This segment is a decreasing function that starts at (-4,5) and ends at an open dot at (-3,2).  There is also a closed dot at (-3,5). The second segment is in the domain \(-3 < x \le -1\).  It starts with an open dot at (-3,2) and increases until approximately (-2.2, 4.8) and then decreases until it ends at a closed dot at (-1,4).  The third segment is in the domain \(-1 < x < \le 2\).  It starts with a closed dot at (-1,-3) has an open dot at (1,-1) and ends at a closed dot at (2,0).  The final segment is in the domain \(x > 2\).  The graph in the segment is an oscillating function that oscillates faster and faster as it approaches x = 2 from the right and the oscillation slows down as it moves away from x=2.
  4. Sketch a graph of a function that satisfies each of the following conditions. \[\mathop {\lim }\limits_{x \to {1^{\, - }}} f\left( x \right) = - 2\hspace{0.75in}\mathop {\lim }\limits_{x \to {1^{\, + }}} f\left( x \right) = 3\hspace{0.75in}f\left( 1 \right) = 6\]
  5. Sketch a graph of a function that satisfies each of the following conditions. \[\mathop {\lim }\limits_{x \to \, - {3^{\, - }}} f\left( x \right) = 1\hspace{0.75in}\mathop {\lim }\limits_{x \to \, - {3^{\, + }}} f\left( x \right) = 1\hspace{0.75in}f\left( { - 3} \right) = 4\]
  6. Sketch a graph of a function that satisfies each of the following conditions. \[\begin{array}{cll}\mathop {\lim }\limits_{x \to \, - {5^{\, - }}} f\left( x \right) = - 1\hspace{0.5in} & \mathop {\lim }\limits_{x \to \, - {5^{\, + }}} f\left( x \right) = 7\hspace{0.5in} & f\left( { - 5} \right) = 4\\ \mathop {\lim }\limits_{x \to 4} f\left( x \right) = 6\hspace{0.5in} & f\left( 4 \right)\,\,\,{\mbox{does not exist}} & \end{array}\]
  7. Explain in your own words what each of the following equations mean. \[\mathop {\lim }\limits_{x \to {8^{\, - }}} f\left( x \right) = 3\hspace{0.75in}\mathop {\lim }\limits_{x \to {8^{\, + }}} f\left( x \right) = - 1\]
  8. Suppose we know that \(\mathop {\lim }\limits_{x \to \, - 7} f\left( x \right) = 18\). If possible, determine the value of \(\mathop {\lim }\limits_{x \to \, - 7{\,^ - }} f\left( x \right)\) and the value of \(\mathop {\lim }\limits_{x \to \, - 7{\,^ + }} f\left( x \right)\). If it is not possible to determine one or both of these values explain why not.
  9. Suppose we know that \(f\left( 6 \right) = - 53\). If possible, determine the value of \(\mathop {\lim }\limits_{x \to \,6{\,^ - }} f\left( x \right)\) and the value of \(\mathop {\lim }\limits_{x \to \,6{\,^ + }} f\left( x \right)\). If it is not possible to determine one or both of these values explain why not.