Stewart Precalc 6e Section 13.2: Finding Limits Algebraically

1 Exercise - Concepts · Level 1

Suppose the following limits exist: \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) and \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)\). Then \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) + g(x)] = \) ____ + ____, and \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) g(x)] = \) ____. These formulas can be stated verbally as follows: The limit of a sum is the ____ of the limits, and the limit of a product is the ____ of the limits.

2 Exercise - Concepts · Level 1

If \(f\) is a polynomial or a rational function and \(a\) is in the domain of \(f\), then \(\operatorname*{lim}\limits_{x \rightarrow a} f(x) = \) ____.

3 Exercise - Skills - Limit Laws · Level 2

Suppose that \(\operatorname*{lim}\limits_{x \rightarrow a} f(x) = -3\), \(\operatorname*{lim}\limits_{x \rightarrow a} g(x) = 0\), and \(\operatorname*{lim}\limits_{x \rightarrow a} h(x) = 8\). Find the value of the given limit. If the limit does not exist, explain why. (a) \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) + h(x)]\) (b) \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x)]^2\) (c) \(\operatorname*{lim}\limits_{x \rightarrow a} \sqrt[3]{h(x)}\) (d) \(\operatorname*{lim}\limits_{x \rightarrow a} \dfrac{1}{f}(x)\) (e) \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)/h(x)\) (f) \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)/f(x)\) (g) \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)/g(x)\) (h) \(\operatorname*{lim}\limits_{x \rightarrow a} \dfrac{2 f(x)}{h(x) - f(x)}\)

4 Exercise - Skills - Limits from Graphs · Level 3

The graphs of \(f\) and \(g\) are given. Use them to evaluate each limit if it exists. If the limit does not exist, explain why. (a) \(\operatorname*{lim}\limits_{x \rightarrow 2} [f(x) + g(x)]\) (b) \(\operatorname*{lim}\limits_{x \rightarrow 1} [f(x) + g(x)]\) (c) \(\operatorname*{lim}\limits_{x \rightarrow 0} [f(x) g(x)]\) (d) \(\operatorname*{lim}\limits_{x \rightarrow -1} f(x)/g(x)\) (e) \(\operatorname*{lim}\limits_{x \rightarrow 2} x^3 f(x)\) (f) \(\operatorname*{lim}\limits_{x \rightarrow 2} \sqrt{3 + f(x)}\)

5 Exercise - Skills - Evaluate Using Limit Laws · Level 1

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow 4} (5 x^2 - 2 x + 3)\)

6 Exercise - Skills - Evaluate Using Limit Laws · Level 1

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow 3} (x^3 + 2)(x^2 - 5 x)\)

7 Exercise - Skills - Evaluate Using Limit Laws · Level 1

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow -1} \dfrac{x - 2}{x^2 + 4 x - 3}\)

8 Exercise - Skills - Evaluate Using Limit Laws · Level 2

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow 1} \left(\dfrac{x^4 + x^2 - 6}{x^4 + 2 x + 3}\right)^2\)

9 Exercise - Skills - Evaluate Using Limit Laws · Level 1

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{t \rightarrow 0} (t + 1)^9 (t^2 - 1)\)

10 Exercise - Skills - Evaluate Using Limit Laws · Level 2

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{u \rightarrow -2} \sqrt{u^4 + 3 u + 6}\)

11 Exercise - Skills - Factoring · Level 2

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 + x - 6}{x - 2}\)

12 Exercise - Skills - Factoring · Level 2

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow -4} \dfrac{x^2 + 5 x + 4}{x^2 + 3 x - 4}\)

13 Exercise - Skills - Direct Substitution · Level 1

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 - x + 6}{x + 2}\)

14 Exercise - Skills - Factoring · Level 2

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{x^3 - 1}{x^2 - 1}\)

15 Exercise - Skills - Factoring · Level 2

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{t \rightarrow -3} \dfrac{t^2 - 9}{2 t^2 + 7 t + 3}\)

16 Exercise - Skills - Rationalization · Level 3

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{h \rightarrow 0} (\sqrt{1 + h} - 1)/h\)

17 Exercise - Skills - Expansion · Level 2

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{h \rightarrow 0} ((2 + h)^3 - 8)/h\)

18 Exercise - Skills - Factoring · Level 2

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^4 - 16}{x - 2}\)

19 Exercise - Skills - Rationalization · Level 3

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 7} \dfrac{\sqrt{x + 2} - 3}{x - 7}\)

20 Exercise - Skills - Simplification · Level 3

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{h \rightarrow 0} ((3 + h)^{-1} - 3^{-1})/h\)

21 Exercise - Skills - Simplification · Level 3

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{x \rightarrow -4} \dfrac{\dfrac{1}{4} + \dfrac{1}{x}}{4 + x}\)

22 Exercise - Skills - Simplification · Level 3

Evaluate the limit if it exists. \(\operatorname*{lim}\limits_{t \rightarrow 0} (\dfrac{1}{t} - 1/(t^2 + t))\)

23 Exercise - Skills - Limits with Graphing Aid · Level 3

Find the limit and use a graphing device to confirm your result graphically. \(\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{x^2 - 1}{\sqrt{x} - 1}\)

24 Exercise - Skills - Limits with Graphing Aid · Level 2

Find the limit and use a graphing device to confirm your result graphically. \(\operatorname*{lim}\limits_{x \rightarrow 0} ((4 + x)^3 - 64)/x\)

25 Exercise - Skills - Limits with Graphing Aid · Level 3

Find the limit and use a graphing device to confirm your result graphically. \(\operatorname*{lim}\limits_{x \rightarrow -1} \dfrac{x^2 - x - 2}{x^3 - x}\)

26 Exercise - Skills - Limits with Graphing Aid · Level 3

Find the limit and use a graphing device to confirm your result graphically. \(\operatorname*{lim}\limits_{x \rightarrow -1} \dfrac{x^8 - 1}{x^5 - x}\)

27 Exercise - Skills - Estimation and Limit Laws · Level 3

(a) Estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow 0} x/(\sqrt{1 + 3 x} - 1)\) by graphing the function \(f(x) = x/(\sqrt{1 + 3 x} - 1)\). (b) Make a table of values of \(f(x)\) for \(x\) close to \(0\), and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct.

28 Exercise - Skills - Estimation and Limit Laws · Level 3

(a) Use a graph of \(f(x) = (\sqrt{3 + x} - \sqrt{3})/x\) to estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x)\) to two decimal places. (b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.

29 Exercise - Skills - Absolute Value Limits · Level 1

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow -4} |x + 4|\)

30 Exercise - Skills - One-Sided Limit with Absolute Value · Level 2

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow -4^-} |x + 4|/(x + 4)\)

31 Exercise - Skills - Absolute Value Limits · Level 2

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 2} |x - 2|/(x - 2)\)

32 Exercise - Skills - Absolute Value Limits · Level 3

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 1.5} (2 x^2 - 3 x)/|2 x - 3|\)

33 Exercise - Skills - One-Sided Limits · Level 3

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 0^-} \left(\dfrac{1}{x} - 1/|x|\right)\)

34 Exercise - Skills - One-Sided Limits · Level 3

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 0^+} \left(\dfrac{1}{x} - 1/|x|\right)\)

35 Exercise - Skills - Piecewise Function · Level 3

Let \(f(x) = \begin{cases} x - 1 & \quad \text{if } x < 2 \\ x^2 - 4 x + 6 & \quad \text{if } x \geq 2 \end{cases}\). (a) Find \(\operatorname*{lim}\limits_{x \rightarrow 2^-} f(x)\) and \(\operatorname*{lim}\limits_{x \rightarrow 2^+} f(x)\). (b) Does \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x)\) exist? (c) Sketch the graph of \(f\).

36 Exercise - Skills - Piecewise Function · Level 3

Let \(h(x) = \begin{cases} x & \quad \text{if } x < 0 \\ x^2 & \quad \text{if } 0 < x \leq 2 \\ 8 - x & \quad \text{if } x > 2 \end{cases}\). (a) Evaluate each limit if it exists: (i) \(\operatorname*{lim}\limits_{x \rightarrow 0^-} h(x)\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow 0} h(x)\) (iii) \(\operatorname*{lim}\limits_{x \rightarrow 1} h(x)\) (iv) \(\operatorname*{lim}\limits_{x \rightarrow 2^-} h(x)\) (v) \(\operatorname*{lim}\limits_{x \rightarrow 2^+} h(x)\) (vi) \(\operatorname*{lim}\limits_{x \rightarrow 2} h(x)\) (b) Sketch the graph of \(h\).

37 Exercise - Discovery, Discussion, Writing · Level 3

Cancellation and Limits. (a) What is wrong with the following equation? \(\dfrac{x^2 + x - 6}{x - 2} = x + 3\) (b) In view of part (a), explain why the equation \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 + x - 6}{x - 2} = \operatorname*{lim}\limits_{x \rightarrow 2} (x + 3)\) is correct.

38 Exercise - Discovery, Discussion, Writing · Level 3

The Lorentz Contraction. In the theory of relativity the Lorentz contraction formula \(L = L_0 \sqrt{1 - v^2/c^2}\) expresses the length \(L\) of an object as a function of its velocity \(v\) with respect to an observer, where \(L_0\) is the length of the object at rest and \(c\) is the speed of light. Find \(\operatorname*{lim}\limits_{v \rightarrow c^-} L\), and interpret the result. Why is a left-hand limit necessary?

39 Exercise - Discovery, Discussion, Writing · Level 4

Limits of Sums and Products. (a) Show by means of an example that \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) + g(x)]\) may exist even though neither \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) nor \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)\) exists. (b) Show by means of an example that \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) g(x)] \) may exist even though neither \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) nor \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)\) exists.

40 Example - Using the Limit Laws · Level 2

Use the Limit Laws and the graphs of \(f\) and \(g\) in Figure 1 to evaluate the following limits if they exist.

(a) \(\operatorname*{lim}\limits_{x \rightarrow -2} [f(x) + 5 g(x)]\)

Answer for 40(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow 1} [f(x) g(x)]\)

Answer for 40(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{f(x)}{g(x)}\)

Answer for 40(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow 1} [f(x)]^3\)

Answer for 40(d)

Enter your answer directly below each part above.

41 Example - Using the Limit Laws · Level 2

Evaluate the following limits, and justify each step.

(a) \(\operatorname*{lim}\limits_{x \rightarrow 5} (2 x^2 - 3 x + 4)\)

Answer for 41(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{x^3 + 2 x^2 - 1}{5 - 3 x}\)

Answer for 41(b)

Enter your answer directly below each part above.

42 Example - Finding Limits by Direct Substitution · Level 1

Evaluate the following limits.

(a) \(\operatorname*{lim}\limits_{x \rightarrow 3} (2 x^3 - 10 x - 8)\)

Answer for 42(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow -1} \dfrac{x^2 + 5 x}{x^4 + 2}\)

Answer for 42(b)

Enter your answer directly below each part above.

43 Example - Finding a Limit by Canceling a Common Factor · Level 2

Find \(\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{x - 1}{x^2 - 1}\).

44 Example - Finding a Limit by Simplifying · Level 2

Evaluate \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(3 + h)^2 - 9}{h}\).

45 Example - Finding a Limit by Rationalizing · Level 3

Find \(\operatorname*{lim}\limits_{t \rightarrow 0} \dfrac{\sqrt{t^2 + 9} - 3}{t^2}\).

46 Example - Comparing Right and Left Limits · Level 2

Show that \(\operatorname*{lim}\limits_{x \rightarrow 0} |x| = 0\).

47 Example - Comparing Right and Left Limits · Level 3

Prove that \(\operatorname*{lim}\limits_{x \rightarrow 0} |x|/x\) does not exist.

48 Example - Limit of Piecewise-Defined Function · Level 3

Let \(f(x) = \begin{cases} \sqrt{x - 4} & \quad \text{if } x > 4 \\ 8 - 2x & \quad \text{if } x < 4 \end{cases}\). Determine whether \(\operatorname*{lim}\limits_{x \rightarrow 4} f(x)\) exists.

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