Stewart 9th Section 3.4: Limits at Infinity; Horizontal Asymptotes

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Stewart 9th Section 3.4: Limits at Infinity; Horizontal Asymptotes 0/89
1 Conceptual · Level 1
Explain in your own words the meaning of each of the following.
(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 5\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = 3\)

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2 Conceptual - asymptotes · Level 1
(a) Can the graph of \(y = f(x)\) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.
(b) How many horizontal asymptotes can the graph of \(y = f(x)\) have? Sketch graphs to illustrate the possibilities.

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3 Reading limits from graph · Level 2
For the function \(f\) whose graph is given, state the following.
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(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x)\)
(c) \(\operatorname*{lim}\limits_{x \rightarrow 1} f(x)\)
(d) \(\operatorname*{lim}\limits_{x \rightarrow 3} f(x)\)
(e) The equations of the asymptotes

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4 Reading limits from graph · Level 2
For the function \(g\) whose graph is given, state the following.
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(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} g(x)\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} g(x)\)
(c) \(\operatorname*{lim}\limits_{x \rightarrow 0} g(x)\)
(d) \(\operatorname*{lim}\limits_{x \rightarrow 2^-} g(x)\)
(e) \(\operatorname*{lim}\limits_{x \rightarrow 2^+} g(x)\)
(f) The equations of the asymptotes

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5 Numerical estimation · Level 2
Guess the value of the limit \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^2}{2^x}\) by evaluating the function \(f(x) = x^2 / 2^x\) for \(x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50\), and \(100\). Then use a graph of \(f\) to support your guess.
6 Numerical estimation · Level 2
(a) Use a graph of \(f(x) = \left(1 - \dfrac{2}{x}\right)^x\) to estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\) correct to two decimal places.
(b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places.

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7 Justified limit · Level 2
Evaluate the limit and justify each step by indicating the appropriate properties of limits. \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{2 x^2 - 7}{5 x^2 + x - 3}\)
8 Justified limit · Level 3
Evaluate the limit and justify each step by indicating the appropriate properties of limits. \(\operatorname*{lim}\limits_{x \rightarrow \infty} \sqrt{\dfrac{9 x^3 + 8 x - 4}{3 - 5 x + x^3}}\)
9 Limit at infinity · Level 1
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{4 x + 3}{5 x - 1} \)
10 Limit at infinity · Level 1
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{-2}{3 x + 7} \)
11 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{t \rightarrow -\infty} \dfrac{3 t^2 + t}{t^3 - 4 t + 1} \)
12 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{t \rightarrow -\infty} \dfrac{6 t^2 + t - 5}{9 - 2 t^2} \)
13 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{r \rightarrow \infty} \dfrac{r - r^3}{2 - r^2 + 3 r^3} \)
14 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{3 x^3 - 8 x + 2}{4 x^3 - 5 x^2 - 2} \)
15 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{4 - \sqrt{x}}{2 + \sqrt{x}} \)
16 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{u \rightarrow -\infty} \dfrac{(u^2 + 1)(2 u^2 - 1)}{(u^2 + 2)^2} \)
17 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sqrt{x + 3 x^2}}{4 x - 1} \)
18 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{t \rightarrow \infty} \dfrac{t + 3}{\sqrt{2 t^2 - 1}} \)
19 Limit at infinity · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sqrt{1 + 4 x^6}}{2 - x^3} \)
20 Limit at infinity · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{\sqrt{1 + 4 x^6}}{2 - x^3} \)
21 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{2 x^5 - x}{x^4 + 3} \)
22 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{q \rightarrow \infty} \dfrac{q^3 + 6 q - 4}{4 q^2 - 3 q + 3} \)
23 Limit at infinity · Level 1
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \cos x \)
24 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1 + x^6}{x^4 + 1} \)
25 Difference of radicals · Level 3
\( \operatorname*{lim}\limits_{t \rightarrow \infty} (\sqrt{25 t^2 + 2} - 5 t) \)
26 Sum involving radicals · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow -\infty} (\sqrt{4 x^2 + 3 x} + 2 x) \)
27 Difference of radicals · Level 3
\( \operatorname*{lim}\limits_{x \rightarrow \infty} (\sqrt{x^2 + a x} - \sqrt{x^2 + b x}) \)
28 Limit at infinity · Level 1
\( \operatorname*{lim}\limits_{x \rightarrow \infty} (x - \sqrt{x}) \)
29 Limit at infinity · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow -\infty} (x^2 + 2 x^7) \)
30 Squeeze theorem · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sin^2 x}{x^2 + 1} \)
31 Substitution limit · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow \infty} x \sin \dfrac{1}{x} \)
32 Substitution limit · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow \infty} \sqrt{x} \sin \dfrac{1}{x} \)
33 Limit estimation and proof · Level 3
(a) Estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow -\infty} (\sqrt{x^2 + x + 1} + x)\) by graphing the function \(f(x) = \sqrt{x^2 + x + 1} + x\).
(b) Use a table of values of \(f(x)\) to guess the value of the limit.
(c) Prove that your guess is correct.

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34 Limit estimation and proof · Level 3
(a) Use a graph of \(f(x) = \sqrt{3 x^2 + 8 x + 6} - \sqrt{3 x^2 + 3 x + 1}\) to estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\) to one decimal place.
(b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places.
(c) Find the exact value of the limit.

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35 Asymptotes · Level 2
Find the horizontal and vertical asymptotes of each curve. \(y = \dfrac{5 + 4 x}{x + 3}\)
36 Asymptotes · Level 2
Find the horizontal and vertical asymptotes of each curve. \(y = \dfrac{2 x^2 + 1}{3 x^2 + 2 x - 1}\)
37 Asymptotes · Level 2
Find the horizontal and vertical asymptotes of each curve. \(y = \dfrac{2 x^2 + x - 1}{x^2 + x - 2}\)
38 Asymptotes · Level 2
Find the horizontal and vertical asymptotes of each curve. \(y = \dfrac{1 + x^4}{x^2 - x^4}\)
39 Asymptotes · Level 3
Find the horizontal and vertical asymptotes of each curve. \(y = \dfrac{x^3 - x}{x^2 - 6 x + 5}\)
40 Asymptotes · Level 3
Find the horizontal and vertical asymptotes of each curve. \(y = \dfrac{x - 9}{\sqrt{4 x^2 + 3 x + 2}}\)
41 Estimating asymptote · Level 3
Estimate the horizontal asymptote of the function \(f(x) = \dfrac{3 x^3 + 500 x^2}{x^3 + 500 x^2 + 100 x + 2000}\) by graphing \(f\) for \(-10 \leq x \leq 10\). Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?
42 Asymptotes by graph · Level 3
(a) Graph the function \(f(x) = \dfrac{\sqrt{2 x^2 + 1}}{3 x - 5}\). How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sqrt{2 x^2 + 1}}{3 x - 5}\) and \(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{\sqrt{2 x^2 + 1}}{3 x - 5}\).
(b) By calculating values of \(f(x)\), give numerical estimates of the limits in part (a).
(c) Calculate the exact values of the limits in part (a).

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43 Polynomial ratios · Level 2
Let \(P\) and \(Q\) be polynomials. Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{P(x)}{Q(x)}\) if the degree of \(P\) is (a) less than the degree of \(Q\) and (b) greater than the degree of \(Q\).
44 Power functions · Level 2
Make a rough sketch of the curve \(y = x^n\) (\(n\) an integer) for the following five cases: (i) \(n = 0\) (ii) \(n > 0\), \(n\) odd (iii) \(n > 0\), \(n\) even (iv) \(n < 0\), \(n\) odd (v) \(n < 0\), \(n\) even Then use these sketches to find the following limits.
(a) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} x^n\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow 0^-} x^n\)
(c) \(\operatorname*{lim}\limits_{x \rightarrow \infty} x^n\)
(d) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} x^n\)

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45 Build a function · Level 3
Find a formula for a function \(f\) that satisfies the following conditions: \(\operatorname*{lim}\limits_{x \rightarrow \pm \infty} f(x) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x) = -\infty\), \(f(2) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow 3^-} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow 3^+} f(x) = -\infty\)
46 Build a function · Level 2
Find a formula for a function that has vertical asymptotes \(x = 1\) and \(x = 3\) and horizontal asymptote \(y = 1\).
47 Build a function · Level 4
A function \(f\) is a ratio of quadratic functions and has a vertical asymptote \(x = 4\) and just one \(x\)-intercept, \(x = 1\). It is known that \(f\) has a removable discontinuity at \(x = -1\) and \(\operatorname*{lim}\limits_{x \rightarrow -1} f(x) = 2\). Evaluate
(a) \(f(0)\)
(b) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\)

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48 Sketch from asymptotes · Level 3
Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. \(y = \dfrac{1 + 2 x^2}{1 + x^2}\)
49 Sketch from asymptotes · Level 3
Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. \(y = \dfrac{1 - x}{1 + x}\)
50 Sketch from asymptotes · Level 3
Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. \(y = \dfrac{x}{\sqrt{x^2 + 1}}\)
51 Sketch from asymptotes · Level 3
Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. \(y = \dfrac{x}{x^2 + 1}\)
52 Sketch using infinity limits · Level 2
Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \(y = 2 x^3 - x^4\)
53 Sketch using infinity limits · Level 2
Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \(y = x^4 - x^6\)
54 Sketch using infinity limits · Level 3
Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \(y = x^3 (x + 2)^2 (x - 1)\)
55 Sketch using infinity limits · Level 3
Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \(y = (3 - x)(1 + x)^2 (1 - x)^4\)
56 Sketch using infinity limits · Level 3
Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \(y = x^2 (x^2 - 1)^2 (x + 2)\)
57 Sketch from conditions · Level 2
Sketch the graph of a function that satisfies all of the given conditions. \(f(2) = 4\), \(f(-2) = -4\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 2\)
58 Sketch from conditions · Level 3
Sketch the graph of a function that satisfies all of the given conditions. \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow -2^-} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow -2^+} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = \infty\)
59 Sketch from conditions · Level 3
Sketch the graph of a function that satisfies all of the given conditions. \(f'(2) = 0\), \(f(2) = -1\), \(f(0) = 0\), \(f'(x) < 0\) if \(0 < x < 2\), \(f'(x) > 0\) if \(x > 2\), \(f''(x) < 0\) if \(0 \leq x < 1\) or if \(x > 4\), \(f''(x) > 0\) if \(1 < x < 4\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 1\), \(f(-x) = f(x)\) for all \(x\)
60 Sketch from conditions · Level 3
Sketch the graph of a function that satisfies all of the given conditions. \(f'(2) = 0\), \(f'(0) = 1\), \(f'(x) > 0\) if \(0 < x < 2\), \(f'(x) < 0\) if \(x > 2\), \(f''(x) < 0\) if \(0 < x < 4\), \(f''(x) > 0\) if \(x > 4\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 0\), \(f(-x) = -f(x)\) for all \(x\)
61 Sketch from conditions · Level 3
Sketch the graph of a function that satisfies all of the given conditions. \(f(1) = f'(1) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow 2^+} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow 2^-} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 0\), \(f''(x) > 0\) for \(x > 2\), \(f''(x) < 0\) for \(x < 0\) and for \(0 < x < 2\)
62 Sketch from conditions · Level 3
Sketch the graph of a function that satisfies all of the given conditions. \(g(0) = 0\), \(g''(x) < 0\) for \(x \neq 0\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} g(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} g(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow 0^-} g'(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow 0^+} g'(x) = \infty\)
63 Squeeze theorem · Level 2
(a) Use the Squeeze Theorem to evaluate \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sin x}{x}\).
(b) Graph \(f(x) = (\sin x)/x\). How many times does the graph cross the asymptote?

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64 End behavior · Level 3
By the end behavior of a function we mean the behavior of its values as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\).
(a) Describe and compare the end behavior of the functions \(P(x) = 3 x^5 - 5 x^3 + 2 x\) and \(Q(x) = 3 x^5\) by graphing both functions in the viewing rectangles \([-2, 2]\) by \([-2, 2]\) and \([-10, 10]\) by \([-10000, 10000]\).
(b) Two functions are said to have the same end behavior if their ratio approaches \(1\) as \(x \rightarrow \infty\). Show that \(P\) and \(Q\) have the same end behavior.

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65 Squeeze theorem application · Level 2
Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\) if \(\dfrac{4 x - 1}{x} < f(x) < \dfrac{4 x^2 + 3 x}{x^2}\) for all \(x > 5\).
66 Application - brine concentration · Level 2
(a) A tank contains \(5000\) L of pure water. Brine that contains \(30\) g of salt per liter of water is pumped into the tank at a rate of \(25\) L/min. Show that the concentration of salt after \(t\) minutes (in grams per liter) is \(C(t) = \dfrac{30 t}{200 + t}\).
(b) What happens to the concentration as \(t \rightarrow \infty\)?

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67 Epsilon-N graph · Level 3
Use a graph to find a number \(N\) such that if \(x > N\) then \(|\dfrac{3 x^2 + 1}{2 x^2 + x + 1} - 1.5| < 0.05\).
68 Epsilon-N illustration · Level 3
For the limit \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1 - 3 x}{\sqrt{x^2 + 1}} = -3\), illustrate Definition 5 by finding values of \(N\) that correspond to \(\epsilon = 0.1\) and \(\epsilon = 0.05\).
69 Epsilon-N illustration · Level 3
For the limit \(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1 - 3 x}{\sqrt{x^2 + 1}} = 3\), illustrate Definition 6 by finding values of \(N\) that correspond to \(\epsilon = 0.1\) and \(\epsilon = 0.05\).
70 Infinite limit illustration · Level 3
For the limit \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{3 x}{\sqrt{x - 3}} = \infty\), illustrate Definition 7 by finding a value of \(N\) that corresponds to \(M = 100\).
71 Epsilon-N proof · Level 3
(a) How large do we have to take \(x\) so that \(1/x^2 < 0.0001\)?
(b) Taking \(r = 2\) in Theorem 4, we have the statement \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1}{x^2} = 0\). Prove this directly using Definition 5.

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72 Epsilon-N proof · Level 3
(a) How large do we have to take \(x\) so that \(\dfrac{1}{\sqrt{x}} < 0.0001\)?
(b) Taking \(r = \dfrac{1}{2}\) in Theorem 4, we have the statement \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1}{\sqrt{x}} = 0\). Prove this directly using Definition 5.

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73 Epsilon-N proof · Level 3
Use Definition 6 to prove that \(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1}{x} = 0\).
74 Infinite limit proof · Level 3
Prove, using Definition 7, that \(\operatorname*{lim}\limits_{x \rightarrow \infty} x^3 = \infty\).
75 Limit transformation · Level 3
(a) Prove that \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = \operatorname*{lim}\limits_{t \rightarrow 0^+} f\left(\dfrac{1}{t}\right)\) and \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = \operatorname*{lim}\limits_{t \rightarrow 0^-} f\left(\dfrac{1}{t}\right)\) assuming that these limits exist.
(b) Use part (a) and Exercise 63 to find \(\operatorname*{lim}\limits_{x \rightarrow 0^+} x \sin \dfrac{1}{x}\).

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76 Precise definition · Level 3
Formulate a precise definition of \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = -\infty\). Then use your definition to prove that \(\operatorname*{lim}\limits_{x \rightarrow -\infty} (1 + x^3) = -\infty\).
77 Example - reading limits from graph · Level 2
Find the infinite limits, limits at infinity, and asymptotes for the function \(f\) whose graph is shown in Figure 5.
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78 Example - limits of 1/x at infinity · Level 1
Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1}{x}\) and \(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1}{x}\).
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79 Example - rational function limit · Level 2
Evaluate the following limit and indicate which properties of limits are used at each stage. \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{3 x^2 - x - 2}{5 x^2 + 4 x + 1}\)
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80 Example - horizontal asymptotes · Level 3
Find the horizontal asymptotes of the graph of the function \(f(x) = \dfrac{\sqrt{2 x^2 + 1}}{3 x - 5}\).
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81 Example - difference of radicals · Level 3
Compute \(\operatorname*{lim}\limits_{x \rightarrow \infty} (\sqrt{x^2 + 1} - x)\).
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82 Example - substitution · Level 2
Evaluate \(\operatorname*{lim}\limits_{x \rightarrow \infty} \sin \dfrac{1}{x}\).
83 Example - oscillating limit · Level 1
Evaluate \(\operatorname*{lim}\limits_{x \rightarrow \infty} \sin x\).
84 Example - infinite limits at infinity · Level 1
Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} x^3\) and \(\operatorname*{lim}\limits_{x \rightarrow -\infty} x^3\).
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85 Example - polynomial at infinity · Level 2
Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} (x^2 - x)\).
86 Example - rational function divergence · Level 2
Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^2 + x}{3 - x}\).
87 Example - rough sketch of polynomial · Level 3
Sketch the graph of \(y = (x - 2)^4 (x + 1)^3 (x - 1)\) by finding its intercepts and its limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\).
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88 Example - epsilon-N from graph · Level 3
Use a graph to find a number \(N\) such that if \(x > N\) then \(|\dfrac{3 x^2 - x - 2}{5 x^2 + 4 x + 1} - 0.6| < 0.1\).
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89 Example - epsilon-N proof · Level 3
Use Definition 5 to prove that \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1}{x} = 0\).
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