Skip to main content
🔗

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)f(x) is shown below. We analyze the behaviour of f(x)f(x) around x=−5, x=−2, x=−1 and x=0, and x=4.

🔗

🔗

🔗
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, so

limx→−5−f(x)=2.

🔗
Similarly, as x gets closer and closer −5 from the right, then f(x) approaches −3, so

limx→−5+f(x)=−3.

🔗
As the right-hand limit and left-hand limit are not equal at −5, we know that

limx→−5f(x) does not exist.

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

limx→−2−f(x)=3.5.

🔗
Similarly, as x gets closer and closer −2 from the right, then f(x) again approaches 3.5, so

limx→−2+f(x)=3.5.

🔗
As the right-hand limit and left-hand limit are both equal to 3.5, we know that

limx→−2f(x)=3.5.

🔗
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.

🔗
Summarizing, we have:

f(−5) is undefinedf(−2)=0f(−1)=−2f(0)=−2limx→−5−f(x)=2limx→−2−f(x)=3.5limx→−1−f(x)=0limx→0−f(x)=−2    limx→−5+f(x)=−3limx→−2+f(x)=3.5limx→−1+f(x)=−2limx→0+f(x)=−2    limx→−5f(x)=DNElimx→−2f(x)=3.5limx→−1f(x)=DNElimx→0f(x)=−2
🔗
Exercises for Section 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
    \(\lim\limits_{x\to 2} f(x) = 2\)
login