Finding Limits Section 2.2 Solutions
Finding Limits section 2.2 Solutions1.1.9-0.1099xf(x)limx 21.99-0.1111.9992-0.1111 ?2.001-0.11112.01-0.11122.1-0.1124x 2 01111 . The actual limit is -1/9x 13 x 2222.-0.1-0.013.993343.999934 sin( x )lim 4.00000x 501.010.12483.-0.10.3997xf(x)-0.010.04008 cos( x ) 8 0.00000x 0xlim4.x 4.9f(x)0.2020limx 54.990.20025.10.1980ln( x) ln 5 0.2000x 55.0.90.1266xf(x)limx 10.990.1252x 4 0.1250 . The actual limit is 1/8x 3 x 2821.10.1235
90.018.98790.17.8333sin(9 x) 8.9999x 0x x 1 7. limdoes not exist.x 1 x 1limFor values of x to the left of 2, lim x 1 x 1 1 , whereasx 1for values of x to the right of 2, lim x 1 x 1 1x 18. lim f ( x) lim( x 2 2) 2x 0x 09. lim sin( x ) 0x 110.2does not exist because the function increasesx 3and decreases without bound as x approaches -3.limx 311. DNE12. DNE13.(a) f(1) exists. The black dot at (1, 2) indicates that f(1) 2.(b)lim f ( x ) does not exist. As x approaches 1 from thex 1left, f(x) approaches 5, whereas as x approaches 1from the right, f(x) approaches 1.(c)(d)f(4) does not exist. The hollow circle at (4, 2) indicates that f is not defined at 4.lim f ( x )x 4exists. As x approaches 4, f(x) approachesf ( x) 2 .2: limx 4
14.(a)(b)f(2) does not exist. The vertical line indicates that f is not defined at 2.lim f ( x ) 2 does not exist. As x approaches 2, the values of f(x) do not approach a(c)(d)specific number.f(0) exists. The red dot at (0, 5) indicates that f(0) 5.lim f ( x ) does not exist. As x approaches 0 from the left, f(x) approaches 3.5, whereas as(e)(f)x approaches 0 from the right, f(x) approaches 5.f(2) does not exist. The hollow circle at (2, 1/2) indicates that f(2) is not defined.lim x 2 f(x) exists. As x approaches 2, f(x) approaches1/2 So lim f ( x ) 1 / 2(g)(h)f(4) exists. The black dot at (4, 3) indicates that f(4) 3.lim f ( x ) does not exist. As x approaches 4, the values of f(x) do not approach a specificx 2x 0x 2x 4number.15.a(ii) is the correct choice: lim f ( x ) exists for all values of c 4.x c16.One possible answer is
Finding limits section 2.21. Consider the following limit. limx 2x 2x 13 x 222a-Complete the table. (Round your answers to four decimal places.)1.9xf(x)1.991.99922.0012.012.1?b-Use the result to estimate the limit.4sin( x)x 0x2. Consider the following limit. lima-Complete the table. (Round your answers to five decimal 10.15.0015.015.11.0011.011.1b-Use the result to estimate the limit.3.Complete the table and use the result to estimate the limit8cos( x) 8limx 0xxf(x)-0.1-0.01-0.0010?ln( x) ln 5x 5x 54. Complete the table and use the result to estimate the limit. limxf(x)4.94.995. Consider the following. limx 14.9995?x 4x 3 x 282Complete the table and use the result to estimate the limit0.90.990.9991xf(x)?
6. Complete the table and use the result to estimate the limit given limx 0xf(x)-0.1-0.01-0.0010?sin(9 x)x0.0010.010.1 x 1 Use the graph to find the limit (if it exists). (If an answer does not exist,x 1 x 17. Consider the following. limwrite DNE.)8. Consider the following. x 2 2 if x 0f ( x) 4 if x 0Use the graph to find the limit below (if it exists). (If an answer does not exist, write DNE.)9. Consider the following.lim sin( x )x 1
Use the graph to find the limit (if it exists). (If an answer does not exist, write DNE.)2x 3 x 310. Consider the following. limUse the graph to find the limit (if it exists). (If an answer does not exist, write DNE.)11. Use the graph to find the limit (if it exists). (If an answer does not exist, write DNE.) lim cosx 01x12. Use the graph to find the limit (if it exists). (If an answer does not exist, write DNE.) lim 2 tan xx /2
13. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. Ifit does not, write DNE.(a) f(1)(b) lim f ( x )x 1(c) f(4)(d) lim f ( x )x 414.Use the graph of the function f to decide whether the value of the given quantity exists. (If an answer does notexist, enter DNE.)
(a)f(2)(b)lim f ( x )x 2(c)f(0)(d)lim f ( x )x 0(e)f(2)(f)lim f ( x )x 2(g)f(4)(h)lim f ( x )x 415. Consider the following. x2 ,x 2 f ( x) 8 2 x, 2 x 4 6,x 4 a-Sketch the graph of f.b-Identify the values of c for which the following limit exists. lim f ( x )x c(i)(ii)(iii)(iv)The limit exists at all points on the graph except where c 2 and c 4.The limit exists at all points on the graph except where c 4The limit exists at all points on the graph except where c 2.The limit exists at all points on the graph.16. Sketch a graph of a function f that satisfies the given values.f(0) is undefinedlim f ( x ) 5 ; lim f ( x ) 3 ; f(3) 7x 0x 3
2.3 Evaluating Limits Analytically and L’hopital rule1-Find the limit.2-Find the limit.3-Find the limit. lim x 7x 24-Find the limit.5-Find the limit of the trigonometric function. lim 5sin xx 6-Find the limit of the trigonometric function. lim 5 tan xx 07Consider the following function and its graph.f(x) xx x2Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enterDNE.)(a) lim f ( x )x 1(b) lim f ( x )x 0(c) Write a simpler function that agrees with the given function at all but one point.8-
(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)x 2 49limx 7 x 7(b)Write a simpler function that agrees with the given function at all but one point.9(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)2 x 2 2 x 24limx 3x 3(b)Write a simpler function that agrees with the given function at all but one point.10(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)x 3 27limx 3 x 3(b)Write a simpler function that agrees with the given function at all but one point. Use agraphing utility to confirm your result.11(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)e2 x 4x ln 2 e x 2lim(b)Write a simpler function that agrees with the given function at all but one point. Use agraphing utility to confirm your result.12(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)limx 3x 3 27x 3(b)Write a simpler function that agrees with the given function at all but one point. Use agraphing utility to confirm your result.
13-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)x 6 6xlimx 014-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)8cos xx /2 cot xlim15-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)limt 0sin 4t3t16-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)limt 0sin t7t17-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)sin 8 tlimt 0t18-Evaluate each limit using l’Hopital rule. Note that if lim f ( x) c then y c is called ax horizontal asymptote.sin 5 xa-. lim 2x 0 x 6 x6 x3b- limx 0 sin x xcos 5 x 1x 0sin 6 xc- lim7x 4x 3 4 xd- limFor instance, here, y -7/4 is a horizontal asymptote 1 e- lim 7 x sin x x
ln 7 x f- lim x 12 x1/2 7x g- lim x x e x2 h- lim x x e 24 x 5 x1/ 4 i- lim 2x 1 x 7x 6 18 j- lim x 16 x 4 x 16 k- lim 8 xsin 6 x x 0 sin16 x l- lim x 0 sin 26 x sin(16 x) 8 x cos(8 x) m- lim x 08 x sin(8 x) n- limcos xx /2 sin( x )226xo- limx 0 cos x 1p- lim (sec x tan x)x / 2 x q- lim tan( x) ln( ) x 11211 6x 7xr- limx 0x6/ xs- lim xx
8/ x 1 t- lim 1 6 ln x x 1u- lim e x ( x 3 x 2 6) x v- lim (x /2w- limx 0e3 x 3 x 1)2 ln( x 1)5 cos x 1x
2.3 Evaluating Limits Analytically and L’hopital rule Solutions1-Find the limit.Ans: 162-Find the limit.Ans: 5123-Find the limit. lim x 7x 2Ans: 34-Find the limit.Ans: 1/25-Find the limit of the trigonometric function. lim 5sin xx Ans: 06-Find the limit of the trigonometric function. lim 5 tan xx 0Ans: 07Consider the following function and its graph.f(x) xx x2
Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enterDNE.)(a) lim f ( x )x 1Ans: DNE(b) lim f ( x )x 0Ans:1(c) Write a simpler function that agrees with the given function at all but one point.1Ans: g ( x) x 18(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)x 2 49x 7 x 7Ans: -14lim(b)Write a simpler function that agrees with the given function at all but one point.Ans: g ( x ) x 79(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)2 x 2 2 x 24x 3x 3Ans: -14lim(b)Write a simpler function that agrees with the given function at all but one point.Ans: g ( x ) 2 x 810(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)
x 3 27x 3 x 3Ans: 27lim(b)Write a simpler function that agrees with the given function at all but one point. Use agraphing utility to confirm your result.Ans: g ( x) x 3 3 x 911(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)e2 x 4limx ln 2 e x 2Ans: 4(b)Write a simpler function that agrees with the given function at all but one point. Use agraphing utility to confirm your result.Ans: g ( x) e x 212(a)Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)x 3 27x 3 x 3Ans: 27lim(b)Write a simpler function that agrees with the given function at all but one point. Use agraphing utility to confirm your result.Ans: g ( x) x 3 3 x 913-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)x 6 6x 0x6Ans:12lim14-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)8 cos xcot xAns: 8limx /2
15-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)sin 4t3tAns: 4/3limt 016-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)sin tt 0 7tAns: 1/7lim17-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)sin 8 tt 0tAns: 0lim18-Evaluate each limit using l’Hopital rule. Note that if lim f ( x) c then y c is called ax horizontal asymptote.sin 5 xa-. lim 2x 0 x 6 xAns:6 x3x 0 sin x xAns:b- limcos 5 x 1x 0sin 6 xc- limAns: 07x 4d- limx 3 4 xAns: -7/4For instance, here, y -7/4 is a horizontal asymptote 1 e- lim 7 x sin x x Ans: 7
ln 7 x f- lim x 12 x1/2 Ans: 7x g- lim x x e Ans: 0 x2 h- lim x x e Ans: 0 24 x 5 x1/ 4 i- lim 2x 1 x 7x 6 Ans: 18 j- lim x 16 x 4 x 16 Ans:k- lim 8 xsin 6 x x 0Ans: sin16 x l- lim x 0 sin 26 x Ans: sin(16 x) 8 x cos(8 x) m- lim x 08 x sin(8 x) Ans: 2n- limx /2cos x sin( x )2
Ans: 16 x2o- limx 0 cos x 1Ans:p- lim (sec x tan x)x / 2Ans: 0 x q- lim tan( x) ln( ) x 11211 Ans:6x 7xx 0xr- limAns:s- lim x 6/ xx Ans: 18/ x 1 t- lim 1 6 ln x x 1Ans:u- lim e x ( x 3 x 2 6) x Ans: 0v- lim (x / 2e3 x 3 x 1)2 ln( x 1)Ans: 35cos x 1x 0xAns: 0w- lim
2.4 Continuity and One-Sided limits1-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x c(b) lim f ( x)x c(c) lim f ( x)x c(ii) Is the function continuous at x 4?2-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x c(b) lim f ( x)x c(c) lim f ( x)x c(ii) Is the function continuous at x -3?3-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)
(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x c(b) lim f ( x)x c(c) lim f ( x)x c(ii) Is the function continuous at x -2?4-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x c
(b) lim f ( x)x c(c) lim f ( x)x c(ii) Is the function continuous at x 1?Use some approximations to determine the limits in each of the following problems: 5-85-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.) lim x 2xx 426-Find the limit (if it exists). (If an answer does not exist, enter DNE.)7-Find the limit (if it exists). (If an answer does not exist, enter DNE.)lim 2 cot xx 8-Find the limit (if it exists).lim ln( x 2)x 29-Find the constant a such that the function is continuous on the entire real line. 4x2 , x 1f ( x) ax 8, x 110-Find the constants a and b such that the function is continuous on the entire real line.x 2 5 f ( x) ax b 2 x 3 5x 3 x2 a2, x a 11-Consider the following function g ( x) x a 6,x a Find the constant a such that the function is continuous on the entire real line. 4sin x, x 0 12-Find the constant a such that the function is continuous on the entire real line. f ( x) x a 9 x, x 013-Find the constant a such that the function is continuous on the entire real number line. 2eax 2,x 4f ( x) 2 ln( x 3) x , x 414-Find all values of c such that f is continuous on (- , ).
7 x 2 , x cf ( x) x c x,15-Use the graph to determine the limit. (If an answer does not exist, enter DNE.) f ( x) 2(a) lim f ( x)x 4(b) lim f ( x)x 416-Consider the following function and graph. f ( x) (a)1x 2lim f ( x)x 2 (b) lim f ( x )x 217-Consider the following function and graph. f ( x) tan x4xx 162
(a)lim f ( x)x 2 (b) lim f ( x)x 218-Consider the following function and graph. f ( x) sec(a) x4lim f ( x)x 2 (b) lim f ( x)x 219‐ Consider the following limit. f ( x) 1x 162a-Complete the table. (Round your answers to two decimal places.) 4.5 4.1 4.01 4.001 4 3.999xf(x)? 3.99 3.9 3.999 3.99 3.5b-Use the table to determine(i) lim f ( x)x 4(ii) lim f ( x)x 4(iii) lim f ( x )x 420‐ Consider the following limit. f ( x) cot x4a-Complete the table. (Round your answers to two decimal places.) 4.5 4.1 4.01 4.001 4xf(x)?b-Use the table to determine(i) lim f ( x)x 4(ii) lim f ( x)x 4(iii) lim f ( x )x 4 3.9 3.5
Note that if f (c) undefined and lim f ( x ) then x c is called a vertical asymptote.x c21‐Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.5If an answer does not exist, enter DNE.) f ( x) 2x22-Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.t 5If an answer does not exist, enter DNE.) g ( x ) 2t 2523‐Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.e 2 xIf an answer does not exist, enter DNE.) f ( x) x 524‐Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.If an answer does not exist, enter DNE.) g(x) xe 5x25‐Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.If an answer does not exist, enter DNE.) f(z) ln(z2 25)26‐Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.If an answer does not exist, enter DNE.) f(x) ln(x 5)27-Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.If an answer does not exist, enter DNE.) f(x) 5 tan(πx)28-Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary.5tIf an answer does not exist, enter DNE.) s (t ) sin(t )129-Find the one-sided limit (if it exists). (If the limit does not exist, enter DNE.) lim 2x 5 ( x 5)30‐Find the one-sided limit (if it exists). (If an answer does not exist, enter DNE.)lim 1 x 2 6x2 x 14x2 4x 38 31-Find the one-sided limit (if it exists). (If the limit does not exist, enter DNE.) lim 7 x 0 x 32-Find the one-sided limit (if it exists). (If an answer does not exist, enter DNE.) lim x 033-Find the one-sided limit (if it exists). (If an answer does not exist, enter DNE.)5sin xlimx ( /2) 5cos x
34‐Find the one-sided limit (if it exists). (If an answer does not exist, enter DNE.) limln(x 2 64) x 8
2.4 Continuity and One-Sided limits solutions1-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x cAns: 3(b) lim f ( x)x cAns:3(c) lim f ( x)x cAns: 3(ii) Is the function continuous at x 4?Ans: Yes2-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x cAns: -3(b) lim f ( x)x cAns:-3
(c) lim f ( x)x cAns: -3(ii) Is the function continuous at x -3?Ans: Yes3-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x cAns: 2(b) lim f ( x)x cAns: 2(c) lim f ( x)x cAns: 3(ii) Is the function continuous at x -2?Ans: No4-Use the graph to determine the limit. (If an answer does not exist, enter DNE.)
(i)Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE.)(a) lim f ( x)x cAns: -2(b) lim f ( x)x cAns: 2(c) lim f ( x)x cAns: 2(ii) Is the function continuous at x 1?Ans: NoUse some approximations to determine the limits in each of the following problems: 5-85-Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)xlim 2x 2x 4Ans: 6-Find the limit (if it exists). (If an answer does not exist, enter DNE.)Ans: 17-Find the limit (if it exists). (If an answer does not exist, enter DNE.)lim 2 cot xx Ans: DNE (b/c left-sided limit is while the right-sided limit is )
8-Find the limit (if it exists).lim ln( x 2)x 2Ans: 9-Find the constant a such that the function is continuous on the entire real line. 4x2 , x 1f ( x) ax 8, x 1Ans: a 1210-Find the constants a and b such that the function is continuous on the entire real line.Ans: a -2, b 111-Consider the following.Find the constant a such that the function is continuous on the entire real line.Ans: a 312-Find the constant a such that the function is continuous on the entire real line. 4sin x, x 0 f ( x) x a 9 x, x 0Ans: a 413Find the constant a such that the function is continuous on the entire real number line. 2eax 2,x 4f ( x) 2 ln( x 3) x , x 4ln 3Ans a 214-Find all values of c such that f is continuous on (- , ).c ;
15-Use the graph to determine the limit. (If an answer does not exist, enter DNE.) f ( x) 2(a) lim f ( x)x 4Ans: (b) lim f ( x)x 4Ans: 16-Consider the following function and graph. f ( x) (a)1x 2lim f ( x)x 2 Ans: (b) lim f ( x )x 2Ans:- 17-Consider the following function and graph. f ( x) tan x4xx 162
(a)lim f ( x)x 2
6. x-0.1 -0.01 -0.001 0.001 0.01 0.1 f(x) 7.8333 8.9879 8.9999 8.9999 8.9879 7.8333 0 sin(9 ) lim 8.9999 x x x 7. 1 x 1 lim x x 1 does not exist. For values of x to the left of 2, 1 x 1 lim 1 x x 1, whereas for values of x to the right of 2, 1 x 1 lim 1 x x 1 8. 2 00 lim ( ) lim( 2) 2
Unit II. Limits (11 days) [SC1] Lab: Computing limits graphically and numerically Informal concept of limit Language of limits, including notation and one-sided limits Calculating limits using algebra [SC7] Properties of limits Limits at infinity and asymptotes Estimating limits num
Capacitors 5 – 6 Fault Finding & Testing Diodes,Varistors, EMC capacitors & Recifiers 7 – 10 Fault Finding & Testing Rotors 11 – 12 Fault Finding & Testing Stators 13 – 14 Fault Finding & Testing DC Welders 15 – 20 Fault Finding & Testing 3 Phase Alternators 21 – 26 Fault Finding & Testing
Lesson 1: Finding Limits Graphically and Numerically When finding limits, you are finding the y-value for what the function is approaching. This can be done in three ways: 1. Make a table 2. Draw a graph 3. Use algebra Limits can fail to exist in three situations: 1. The left-limit is
1.2 Finding Limits Graphically and Numerically Homework on the Web Calculus Home Page Class Notes: Prof. G. Battaly How do you Find Limits? 1. Numerically use only to show how the values are changing x 1.9 1.99 1.999 2.001 2.01 2.1 f(x) 1.2 Finding Limits Graphically and Numerically indeterminant
REVIEW – 1 WEEK Summer Packet Review LIMITS – 2 WEEKS Finding Limits Graphically and Numerically Evaluating Limits Analytically Continuity and One-Sided limits Infinite Limits Limits at Infinity NJSLS Standard(s) Addressed in this unit F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically .
Aug 26, 2010 · Limits of functions o An intuitive understanding of the limiting process. Language of limits, including notation and one-sided limits. Calculating limits using algebra. Properties of limits. Estimating limits from graphs or tables of data. Estimating limits numerically and
m g e t h o d o f c l e a r i n g 200.03 200.03 a a 1-1 8 1-1 8 part section d-d part section c-c part section b-b clearing limits clearing limits slope stake line slope stake line r/w r/w r/w r/w r/w r/w clearing limits clearing limits clearing limits c c c c c f f f f f * 1 0 ' 5 ' 6 ' * 1 0 ' e.o.p. berm ditch 10' slope stake point const. limit
2 15 ' 1 2 15 ' 2 15 ' notes: mainline 1 typ. lighting limits interchange 215' lighting limits interchange lighting limits interchange lighting limits interchange lighting limits interchange 3. 2. 1. interchange lighting limits n.t.s. reports indicates. of what that the lighting justification the mainline will be illuminated regardless
