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Mathematics LibreTexts

1.1E: Exercises

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    11018
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    Exercise \(\PageIndex{1}\)

    Terms and Concepts

    1. In your own words, what does it mean to "find the limit of \(f(x)\) as \(x\) approaches 3"?
    2. An expression of the form \(\frac{0}{0}\) is called _____.
    3. T/F: The limit of \(f(x)\) as \(x\) approaches 5 is \(f(5)\).
    4. Describe three situations where \(\lim\limits_{x\to c}f(x)\) does not exist.

    Exercise \(\PageIndex{2}\)

    For exercises 1 - 2, consider the function \(f(x)=\dfrac{x^2−1}{|x−1|}\).

    1) [T] Complete the following table for the function. Round your solutions to four decimal places.

    \(x\) \(f(x)\) \(x\) \(f(x)\)
    0.9 a. 1.1 e.
    0.99 b. 1.01 f.
    0.999 c. 1.001 g.
    0.9999 d. 1.0001 h.

    2) What do your results in the preceding exercise indicate about the two-sided limit \(\displaystyle \lim_{x→1}f(x)\)? Explain your response.

    Answer

    \(\displaystyle \lim_{x \to 1}f(x)\) does not exist because \(\displaystyle \lim_{x \to 1^−}f(x)=−2≠\lim_{x \to 1^+}f(x)=2\).

    Exercise \(\PageIndex{3}\)

    Consider the function \(f(x)=\left(1+\dfrac{1}{x}\right)^x\). Make a table showing \(f(x)\) for \(x=1,2,3, .....\) . Round your solutions to five decimal places. What can you say about the value of the function \(f(x)\) as \(x\) increases indefinitely?

     
    Answer

    \(\lim_{x \to \infty}\left(1+\dfrac{1}{x}\right)^x=2.7183=e.\)