Stewart 9th Section 1.6: Calculating Limits Using the Limit Laws

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Stewart 9th Section 1.6: Calculating Limits Using the Limit Laws 0/68
1 Limit Laws - Application · Level 2
Given that \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) = 4\), \(\operatorname*{lim}\limits_{x \rightarrow 2} g(x) = -2\), \(\operatorname*{lim}\limits_{x \rightarrow 2} h(x) = 0\), find the limits that exist. If the limit does not exist, explain why.
(a) \(\operatorname*{lim}\limits_{x \rightarrow 2} [f(x) + 5 g(x)]\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow 2} [g(x)]^3\)
(c) \(\operatorname*{lim}\limits_{x \rightarrow 2} \sqrt{f(x)}\)
(d) \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{3 f(x)}{g(x)}\)
(e) \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{g(x)}{h(x)}\)
(f) \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{g(x) h(x)}{f(x)}\)

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2 Limit Laws - Application · Level 2
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.
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(a) \(\operatorname*{lim}\limits_{x \rightarrow 2} [f(x) + g(x)]\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow 0} [f(x) - g(x)]\)
(c) \(\operatorname*{lim}\limits_{x \rightarrow -1} [f(x) g(x)]\)
(d) \(\operatorname*{lim}\limits_{x \rightarrow 3} \dfrac{f(x)}{g(x)}\)
(e) \(\operatorname*{lim}\limits_{x \rightarrow 2} [x^2 f(x)]\)
(f) \(f(-1) + \operatorname*{lim}\limits_{x \rightarrow -1} g(x)\)

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3 Limit Laws - Direct Evaluation · Level 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow 5} (4 x^2 - 5 x)\)
4 Limit Laws - Direct Evaluation · Level 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow -3} (2 x^3 + 6 x^2 - 9)\)
5 Limit Laws - Direct Evaluation · Level 2
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{v \rightarrow 2} (v^2 + 2 v)(2 v^3 - 5)\)
6 Limit Laws - Direct Evaluation · Level 2
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{t \rightarrow -7} \dfrac{3 t^2 + 1}{t^2 - 5 t + 2}\)
7 Limit Laws - Direct Evaluation · Level 2
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{u \rightarrow -2} \sqrt{9 - u^3 + 2 u^2}\)
8 Limit Laws - Direct Evaluation · Level 2
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{x \rightarrow 3} \sqrt[3]{x + 5} (2 x^2 - 3 x)\)
9 Limit Laws - Direct Evaluation · Level 2
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{t \rightarrow -1} \left(\dfrac{2 t^5 - t^3}{5 t^2 + 4}\right)^3\)
10 Limit Laws - Algebraic · Level 2
(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.

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11 Limit Laws - Algebraic · Level 1
\( \operatorname*{lim}\limits_{x \rightarrow -2} (3 x - 7) \)
12 Limit Laws - Algebraic · Level 1
\( \operatorname*{lim}\limits_{x \rightarrow 6} \left(8 - \dfrac{1}{2} x\right) \)
13 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{t \rightarrow 4} \dfrac{t^2 - 2 t - 8}{t - 4} \)
14 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow -3} \dfrac{x^2 + 3 x}{x^2 - x - 12} \)
15 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 + 5 x + 4}{x - 2} \)
16 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow 4} \dfrac{x^2 + 3 x}{x^2 - x - 12} \)
17 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{x^2 - x - 6}{3 x^2 + 5 x - 2} \)
18 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow -5} \dfrac{2 x^2 + 9 x - 5}{x^2 - 25} \)
19 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{t \rightarrow 3} \dfrac{t^3 - 27}{t^2 - 9} \)
20 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{u \rightarrow -1} \dfrac{u + 1}{u^3 + 1} \)
21 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(h - 3)^2 - 9}{h} \)
22 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow 9} \dfrac{9 - x}{3 - \sqrt{x}} \)
23 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\sqrt{9 + h} - 3}{h} \)
24 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{2 - x}{\sqrt{x + 2} - 2} \)
25 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow 3} \dfrac{\dfrac{1}{x} - \dfrac{1}{3}}{x - 3} \)
26 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(-2 + h)^{-1} + 2^{-1}}{h} \)
27 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{t \rightarrow 0} \dfrac{\sqrt{1 + t} - \sqrt{1 - t}}{t} \)
28 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{t \rightarrow 0} \left(\dfrac{1}{t} - \dfrac{1}{t^2 + t}\right) \)
29 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow 16} \dfrac{4 - \sqrt{x}}{16 x - x^2} \)
30 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 - 4 x + 4}{x^4 - 3 x^2 - 4} \)
31 Limit Laws - Algebraic · Level 4
\( \operatorname*{lim}\limits_{t \rightarrow 0} \left(\dfrac{1}{t \sqrt{1 + t}} - \dfrac{1}{t}\right) \)
32 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow -4} \dfrac{\sqrt{x^2 + 9} - 5}{x + 4} \)
33 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(x + h)^3 - x^3}{h} \)
34 Limit Laws - Algebraic · Level 4
\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\dfrac{1}{(x + h)^2} - \dfrac{1}{x^2}}{h} \)
35 Limit Laws - Graphical/Numerical · Level 3
(a) Estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{x}{\sqrt{1 + 3 x} - 1}\) by graphing the function \(f(x) = \dfrac{x}{\sqrt{1 + 3 x} - 1}\).
(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.

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36 Limit Laws - Graphical/Numerical · Level 3
(a) Use a graph of \(f(x) = \dfrac{\sqrt{3 + x} - \sqrt{3}}{x}\) 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.

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37 Limit Laws - Squeeze Theorem · Level 3
Use the Squeeze Theorem to show that \(\operatorname*{lim}\limits_{x \rightarrow 0} x^2 \cos(20 \pi x) = 0\). Illustrate by graphing the functions \(f(x) = -x^2\), \(g(x) = x^2 \cos(20 \pi x)\), and \(h(x) = x^2\) on the same screen.
38 Limit Laws - Squeeze Theorem · Level 3
Use the Squeeze Theorem to show that \(\operatorname*{lim}\limits_{x \rightarrow 0} \sqrt{x^3 + x^2} \sin\left(\dfrac{\pi}{x}\right) = 0\). Illustrate by graphing the functions \(f\), \(g\), \(h\) on the same screen.
39 Limit Laws - Squeeze Theorem · Level 2
If \(4 x - 9 \leq f(x) \leq x^2 - 4 x + 7\) for \(x \geq 0\), find \(\operatorname*{lim}\limits_{x \rightarrow 4} f(x)\).
40 Limit Laws - Squeeze Theorem · Level 2
If \(2 x \leq g(x) \leq x^4 - x^2 + 2\) for all \(x\), evaluate \(\operatorname*{lim}\limits_{x \rightarrow 1} g(x)\).
41 Limit Laws - Squeeze Theorem · Level 3
Prove that \(\operatorname*{lim}\limits_{x \rightarrow 0} x^4 \cos\left(\dfrac{2}{x}\right) = 0\).
42 Limit Laws - Squeeze Theorem · Level 3
Prove that \(\operatorname*{lim}\limits_{x \rightarrow 0^+} \sqrt{x} [1 + \sin^2\left(\dfrac{2 \pi}{x}\right)] = 0\).
43 Limit Laws - Absolute Value · Level 2
Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow -4} (|x + 4| - 2 x)\)
44 Limit Laws - Absolute Value · Level 3
Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow -4} \dfrac{|x + 4|}{2 x + 8}\)
45 Limit Laws - Absolute Value · Level 3
Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 0.5^-} \dfrac{2 x - 1}{|2 x^3 - x^2|}\)
46 Limit Laws - Absolute Value · Level 3
Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{2 - |x|}{2 + x}\)
47 Limit Laws - Absolute Value · Level 3
Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 0^-} \left(\dfrac{1}{x} - \dfrac{1}{|x|}\right)\)
48 Limit Laws - Absolute Value · Level 3
Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 0^+} \left(\dfrac{1}{x} - \dfrac{1}{|x|}\right)\)
49 Limit Laws - Piecewise/Special · Level 3
The signum (or sign) function, \(\text{sgn} x\), is defined by \(\text{sgn} x = \begin{cases} -1 & \quad \text{if } x < 0 \\ 0 & \quad \text{if } x = 0 \\ 1 & \quad \text{if } x > 0 \end{cases}\)
(a) Sketch the graph of this function.
(b) Find each of the following limits, or explain why it does not exist. (i) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} \text{sgn} x\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow 0^-} \text{sgn} x\) (iii) \(\operatorname*{lim}\limits_{x \rightarrow 0} \text{sgn} x\) (iv) \(\operatorname*{lim}\limits_{x \rightarrow 0} |\text{sgn} x|\)

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50 Limit Laws - Piecewise/Special · Level 3
Let \(g(x) = \text{sgn}(\sin x)\).
(a) Find each of the following limits, or explain why it does not exist. (i) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} g(x)\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow 0^-} g(x)\) (iii) \(\operatorname*{lim}\limits_{x \rightarrow 0} g(x)\) (iv) \(\operatorname*{lim}\limits_{x \rightarrow \pi^-} g(x)\) (v) \(\operatorname*{lim}\limits_{x \rightarrow \pi^+} g(x)\) (vi) \(\operatorname*{lim}\limits_{x \rightarrow \pi} g(x)\)
(b) For which values of \(a\) does \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)\) not exist?
(c) Sketch a graph of \(g\).

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51 Limit Laws - Piecewise/Special · Level 3
Let \(g(x) = \dfrac{x^2 + x - 6}{|x - 2|}\).
(a) Find (i) \(\operatorname*{lim}\limits_{x \rightarrow 2^+} g(x)\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow 2^-} g(x)\)
(b) Does \(\operatorname*{lim}\limits_{x \rightarrow 2} g(x)\) exist?
(c) Sketch the graph of \(g\).

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

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53 Limit Laws - Piecewise/Special · Level 3
Let \(B(t) = \begin{cases} 4 - \dfrac{1}{2} t & \quad \text{if } t < 2 \\ \sqrt{t + c} & \quad \text{if } t \geq 2 \end{cases}\) Find the value of \(c\) so that \(\operatorname*{lim}\limits_{t \rightarrow 2} B(t)\) exists.
54 Limit Laws - Piecewise/Special · Level 3
Let \(g(x) = \begin{cases} x & \quad \text{if } x < 1 \\ 3 & \quad \text{if } x = 1 \\ 2 - x^2 & \quad \text{if } 1 < x \leq 2 \\ x - 3 & \quad \text{if } x > 2 \end{cases}\)
(a) Evaluate each of the following, if it exists. (i) \(\operatorname*{lim}\limits_{x \rightarrow 1^-} g(x)\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow 1^+} g(x)\) (iii) \(g(1)\) (iv) \(\operatorname*{lim}\limits_{x \rightarrow 2^-} g(x)\) (v) \(\operatorname*{lim}\limits_{x \rightarrow 2^+} g(x)\) (vi) \(\operatorname*{lim}\limits_{x \rightarrow 2} g(x)\)
(b) Sketch the graph of \(g\).

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55 Limit Laws - Piecewise/Special · Level 3
(a) If \(\lfloor x \rfloor\) denotes the greatest integer function, evaluate: (i) \(\operatorname*{lim}\limits_{x \rightarrow -2^+} \lfloor x \rfloor\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow -2} \lfloor x \rfloor\) (iii) \(\operatorname*{lim}\limits_{x \rightarrow -2.4} \lfloor x \rfloor\)
(b) If \(n\) is an integer, evaluate: (i) \(\operatorname*{lim}\limits_{x \rightarrow n^-} \lfloor x \rfloor\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow n^+} \lfloor x \rfloor\)
(c) For what values of \(a\) does \(\operatorname*{lim}\limits_{x \rightarrow a} \lfloor x \rfloor\) exist?

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56 Limit Laws - Piecewise/Special · Level 3
Let \(f(x) = \lfloor \cos x \rfloor\), \(-\pi \leq x \leq \pi\).
(a) Sketch the graph of \(f\).
(b) Evaluate each limit, if it exists. (i) \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x)\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{2}^-} f(x)\) (iii) \(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{2}^+} f(x)\) (iv) \(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{2}} f(x)\)
(c) For what values of \(a\) in \((-\pi, \pi)\) does \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) exist?

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57 Limit Laws - Piecewise/Special · Level 3
If \(f(x) = \lfloor x \rfloor + \lfloor -x \rfloor\), show that \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x)\) exists but is not equal to \(f(2)\).
58 Limit Laws - Applied/Proof · Level 3
In the theory of relativity, the Lorentz contraction formula \(L = L_0 \sqrt{1 - \dfrac{v^2}{c^2}}\) 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?
59 Limit Laws - Applied/Proof · Level 3
If \(p\) is a polynomial, show that \(\operatorname*{lim}\limits_{x \rightarrow a} p(x) = p(a)\).
60 Limit Laws - Applied/Proof · Level 3
If \(r\) is a rational function, use Exercise 59 to show that \(\operatorname*{lim}\limits_{x \rightarrow a} r(x) = r(a)\) for every number \(a\) in the domain of \(r\).
61 Limit Laws - Applied/Proof · Level 3
If \(\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{f(x) - 8}{x - 1} = 10\), find \(\operatorname*{lim}\limits_{x \rightarrow 1} f(x)\).
62 Limit Laws - Applied/Proof · Level 3
If \(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{f(x)}{x^2} = 5\), find the following limits.
(a) \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x)\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{f(x)}{x}\)

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63 Limit Laws - Applied/Proof · Level 4
If \(f(x) = \begin{cases} x^2 \text{if x is rational} \\ 0 \text{if x is irrational} \end{cases}\), prove that \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x) = 0\).
64 Limit Laws - Applied/Proof · Level 3
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.
65 Limit Laws - Applied/Proof · Level 3
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.
66 Limit Laws - Applied/Proof · Level 4
Evaluate \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{\sqrt{6 - x} - 2}{\sqrt{3 - x} - 1}\).
67 Limit Laws - Applied/Proof · Level 4
Is there a number \(a\) such that \(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{3 x^2 + a x + a + 3}{x^2 + x - 2}\) exists? If so, find the value of \(a\) and the value of the limit.
68 Limit Laws - Applied/Proof · Level 5
The figure shows a fixed circle \(C_1\) with equation \((x - 1)^2 + y^2 = 1\) and a shrinking circle \(C_2\) with radius \(r\) and center the origin. \(P\) is the point \((0, r)\), \(Q\) is the upper point of intersection of the two circles, and \(R\) is the point of intersection of the line \(P Q\) and the \(x\)-axis. What happens to \(R\) as \(C_2\) shrinks, that is, as \(r \rightarrow 0^+\)?
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