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Solutions for Finding Limits: Numerical and Graphical Approaches

Solutions to Try Its

1. a=5a=5, f(x)=2x24f\left(x\right)=2{x}^{2}-4, and L=46L=46. 2. a. 0; b. 2; c. does not exist; d. 2-2; e. 0; f. does not exist; g. 4; h. 4; i. 4 3. limx0(20sin(x)4x)=5\underset{x\to 0}{\mathrm{lim}}\left(\frac{20\sin \left(x\right)}{4x}\right)=5 Table showing that f(x) approaches 5 from either side as x approaches 0 from either side. 4. does not exist

Solutions to Odd-Numbered Exercises

1. The value of the function, the output, at x=ax=a is f(a)f\left(a\right). When the limxaf(x)\underset{x\to a}{\mathrm{lim}}f\left(x\right) is taken, the values of xx get infinitely close to aa but never equal aa. As the values of xx approach aa from the left and right, the limit is the value that the function is approaching. 3. –4 5. –4 7. 2 9. does not exist 11. 4 13. does not exist 15. Answers will vary. 17. Answers will vary. 19. Answers will vary. 21. Answers will vary. 23. 7.38906 25. 54.59815 27. e6403.428794{e}^{6}\approx 403.428794, e71096.633158{e}^{7}\approx 1096.633158, en{e}^{n} 29. limx2f(x)=1\underset{x\to -2}{\mathrm{lim}}f\left(x\right)=1 31. limx3(x2x6x29)=560.83\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}-x - 6}{{x}^{2}-9}\right)=\frac{5}{6}\approx 0.83 33. limx1(x21x23x+2)=2.00\underset{x\to 1}{\mathrm{lim}}\left(\frac{{x}^{2}-1}{{x}^{2}-3x+2}\right)=-2.00 35. limx1(1010x2x23x+2)=20.00\underset{x\to 1}{\mathrm{lim}}\left(\frac{10 - 10{x}^{2}}{{x}^{2}-3x+2}\right)=20.00 37. limx12(x4x2+4x+1)\underset{x\to \frac{-1}{2}}{\mathrm{lim}}\left(\frac{x}{4{x}^{2}+4x+1}\right) does not exist. Function values decrease without bound as xx approaches –0.5 from either left or right. 39. limx07tanx3x=73\underset{x\to 0}{\mathrm{lim}}\frac{7\tan x}{3x}=\frac{7}{3} Table shows as the function approaches 0, the value is 7 over 3 but the function is undefined at 0. 41. limx02sinx4tanx=12\underset{x\to 0}{\mathrm{lim}}\frac{2\sin x}{4\tan x}=\frac{1}{2} Table shows as the function approaches 0, the value is 1 over 2, but the function is undefined at 0. 43. limx0ee 1x2=1.0\underset{x\to 0}{\mathrm{lim}}{e}^{{e}^{-\text{ }\frac{1}{{x}^{2}}}}=1.0 45. limx1x+1x+1=(x+1)(x+1)=1\underset{x\to -{1}^{-}}{\mathrm{lim}}\frac{|x+1|}{x+1}=\frac{-\left(x+1\right)}{\left(x+1\right)}=-1 and limx1+x+1x+1=(x+1)(x+1)=1\underset{x\to -{1}^{+}}{\mathrm{lim}}\frac{|x+1|}{x+1}=\frac{\left(x+1\right)}{\left(x+1\right)}=1; since the right-hand limit does not equal the left-hand limit, limx1x+1x+1\underset{x\to -1}{\mathrm{lim}}\frac{|x+1|}{x+1} does not exist. 47. limx11(x+1)2\underset{x\to -1}{\mathrm{lim}}\frac{1}{{\left(x+1\right)}^{2}} does not exist. The function increases without bound as xx approaches 1-1 from either side. 49. limx051e2x\underset{x\to 0}{\mathrm{lim}}\frac{5}{1-{e}^{\frac{2}{x}}} does not exist. Function values approach 5 from the left and approach 0 from the right. 51. Through examination of the postulates and an understanding of relativistic physics, as vcv\to c, mm\to \infty . Take this one step further to the solution,

limvcm=limvcmo1(v2/c2)=\underset{v\to {c}^{-}}{\mathrm{lim}}m=\underset{v\to {c}^{-}}{\mathrm{lim}}\frac{{m}_{o}}{\sqrt{1-\left({v}^{2}/{c}^{2}\right)}}=\infty

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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.