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Section 3.3 Computing Limits: Graphically

In this section we look at an example to illustrate the concept of a limit graphically.

The graph of a function \(f(x)\) is shown below. We analyze the behaviour of \(f(x)\) around \(x=-5\text{,}\) \(x=-2\text{,}\) \(x=-1\) and \(x=0\text{,}\) and \(x=4\text{.}\)

Observe that \(f(x)\) is indeed a function (it passes the Vertical Line Test). We now analyze the function at each point separately.

x=-5: Observe that at \(x=-5\) there is no closed circle, thus \(f(-5)\) is undefined. From the graph we see that as \(x\) gets closer and closer to \(-5\) from the left, then \(f(x)\) approaches \(2\text{,}\) so

\begin{equation*} \lim_{x\to -5^-}f(x)=2\text{.} \end{equation*}

Similarly, as \(x\) gets closer and closer \(-5\) from the right, then \(f(x)\) approaches \(-3\text{,}\) so

\begin{equation*} \lim_{x\to -5^+}f(x)=-3\text{.} \end{equation*}

As the right-hand limit and left-hand limit are not equal at \(-5\text{,}\) we know that

\begin{equation*} \lim_{x\to -5}f(x)\mbox{ does not exist.} \end{equation*}

x=-2: Observe that at \(x=-2\) there is a closed circle at \(0\text{,}\) thus \(f(-2)=0\text{.}\) From the graph we see that as \(x\) gets closer and closer to \(-2\) from the left, then \(f(x)\) approaches \(3.5\text{,}\) so

\begin{equation*} \lim_{x\to -2^-}f(x)=3.5\text{.} \end{equation*}

Similarly, as \(x\) gets closer and closer \(-2\) from the right, then \(f(x)\) again approaches \(3.5\text{,}\) so

\begin{equation*} \lim_{x\to -2^+}f(x)=3.5\text{.} \end{equation*}

As the right-hand limit and left-hand limit are both equal to \(3.5\text{,}\) we know that

\begin{equation*} \lim_{x\to -2}f(x)=3.5\text{.} \end{equation*}

Do not be concerned that the limit does not equal 0. This is a discontinuity, which is completely valid, and will be discussed in a later section.

We leave it to the reader to analyze the behaviour of \(f(x)\) for \(x\) close to \(-1\) and \(0\text{.}\)

Summarizing, we have:

\begin{equation*} \begin{array}{cccc} f(-5) \mbox{ is undefined} \amp f(-2)=0 \amp f(-1)=-2 \amp f(0)=-2 \\ \hline \ds{\lim_{x\to -5^-}f(x)=2} \amp \ds{\lim_{x\to -2^-}f(x)=3.5} \amp \ds{\lim_{x\to -1^-}f(x)=0} \amp \ds{\lim_{x\to 0^-}f(x)=-2} \\ ~\amp ~\amp ~\amp ~\\ \ds{\lim_{x\to -5^+}f(x)=-3} \amp \ds{\lim_{x\to -2^+}f(x)=3.5} \amp \ds{\lim_{x\to -1^+}f(x)=-2} \amp \ds{\lim_{x\to 0^+}f(x)=-2}\\ ~\amp ~\amp ~\amp ~\\ \ds{\lim_{x\to -5}f(x)=\mbox{DNE} } \amp \ds{\lim_{x\to -2}f(x)=3.5} \amp \ds{\lim_{x\to -1}f(x)=\mbox{DNE} } \amp \ds{\lim_{x\to 0}f(x)=-2} \end{array} \end{equation*}
Exercises for Section聽3.3.

Evaluate the expressions by reference to this graph:

  1. \(\ds \lim_{x\to 4} f(x)\)

    Answer
    \(\lim\limits_{x \to 4} f(x) = 8\)

  2. \(\ds \lim_{x\to -3} f(x)\)

    Answer
    \(\lim\limits_{x \to -3} f(x) = 6\)

  3. \(\ds \lim_{x\to 0} f(x)\)

    Answer
    \(\lim\limits_{x \to 0} f(x)\) DNE, since the left and right side limits are not the same.

  4. \(\ds \lim_{x\to 0^-} f(x)\)

    Answer
    \(\lim\limits_{x \to 0^{-}} f(x) = -2\text{.}\) Note that this is not equal to \(f(0)\text{,}\) but is instead the value which \(f\) is approaching as \(x \to 0\) from the left.

  5. \(\ds \lim_{x\to 0^+} f(x)\)

    Answer
    \(\lim\limits_{x \to 0^{+}} f(x) = -1\)

  6. \(\ds f(-2)\)

    Answer
    \(f(-2) = 8\)

  7. \(\ds \lim_{x\to 2^-} f(x)\)

    Answer
    \(\lim\limits_{x \to 2^{-}} f(x) = 7\)

  8. \(\ds \lim_{x\to -2^-} f(x)\)

    Answer
    \(\lim\limits_{x \to -2^{-}} f(x) = 6\)

  9. \(\ds \lim_{x\to 0} f(x+1)\)

    Answer
    \(\lim\limits_{x \to 0} f(x+1) = \lim\limits_{x \to 1} f(x) = 3\)

  10. \(\ds f(0)\)

    Answer
    \(f(0) = -1.5\)

  11. \(\ds \lim_{x\to 1^-} f(x-4)\)

    Answer
    \(\lim\limits_{x \to 1^{-}} f(x-4) = \lim\limits_{x \to -3^{-}} f(x) = 6\)

  12. \(\ds \lim_{x\to 0^+} f(x-2)\)

    Answer
    \(\lim\limits_{x \to 0^{+}} f(x-2) = \lim\limits_{x \to -2^{+}} f(x) = 2\)

Evaluate the expressions by reference to this graph:

  1. \(\lim\limits_{x\to -1} f(x)\)

    Answer
    \(\lim\limits_{x\to -1} f(x) =\) DNE, since the left and right-side limits are not equal.

  2. \(\lim\limits_{x\to 0^+} f(x)\)

    Answer
    \(\lim\limits_{x\to 0^+} f(x) = 2\)

  3. \(\lim\limits_{x\to 1} f(x)\)

    Answer
    \(\lim\limits_{x\to 0^+} f(x) = 2\)

  4. \(f(1)\)

    Answer
    \(f(1) = 0\)

  5. \(\lim\limits_{x\to 2} f(x)\)

    Answer
    \(\lim\limits_{x\to 2} f(x) = 2\)
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