You will find a graphing calculator useful for Exercises

Problem 90CE In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit.

In Exercises 5 and 6, explain why the limits do not exist. \(\lim _{x \rightarrow 0} \frac{x}{|x|}\) Equation Transcription: Text Transcription: Lim right arrow 0 x/absolute x

Which of the following statements about the function y = f(x) graphed here are true, and which are false? 1. 2. 3. 4. 5. 6. 7. Image Text Transcription: Which of the following statements about the function y = f(x) graphed here are true, and which are false? x0f(x) exists. x0f(x) = 0 x0f(x) = 1 x1f(x) = 1 x1f(x) = 0 xcf(x) exists at every point c in (-1, 1) x1f(x) does ot exist.

For the function \(g(x)\) graphed here, find the following limits or explain why they do not exist. 1. \(\lim _{x \rightarrow 1} g(x)\) 2. \(lim _{x \rightarrow 2} g(x)\) 3. \(lim _{x \rightarrow 3} g(x)\) 4. \(lim _{x \rightarrow 2.5} g(x)\) Equation Transcription: Text Transcription: Lim x right arrow 1 g(x) Lim x right arrow 2 g(x) Lim x right arrow 2 g(x) Lim x right arrow 2.5 g(x)

For the function ƒ(t) graphed here, find the following limits or explain why they do not exist. 1. \(\lim _{t \rightarrow-2} f(t)\) 2. \(\lim _{t \rightarrow-1} f(t)\) 3. \(\lim _{t \rightarrow 0} f(t)\) 4. \(\lim _{t \rightarrow 1} f(t)\) Equation Transcription: Text Transcription: Lim t right arrow -2 f(t) Lim t right arrow -1 f(t) Lim t right arrow 0 f(t) Lim t right arrow 0.5 f(t)

Which of the following statements about the function \(y=ƒ(x)\) graphed here are true, and which are false? 1. \(\lim _{x \rightarrow 2} f(x) \text { does not exist. }\) 2. \(\lim _{x \rightarrow 2} f(x)=2\) 3. \(\lim _{x \rightarrow 1} f(x) \text { does not exist. }\) 4. \(\lim _{x \rightarrow c} f(x) \text { exists at every point c in }(-1,1) \text {. }\) 5. \(\lim _{x \rightarrow c} f(x) \text { exists at every point cin }(1,3) \text {. }\) Equation Transcription: Text Transcription: Lim x right arrow 2 f(x) Lim x right arrow 2 f(x) Lim x right arrow 1 f(x) Lim x right arrow c f(x) Lim x right arrow c f(x)

Problem 6E In Exercises 5 and 6, explain why the limits do not exist.

Problem 7E Suppose that a function ƒ(x) is defined for all real values of x except x = c. Can anything be said about the existence of Give reasons for your answer.

Problem 8E Suppose that a function ƒ(x) is defined for all x in [-1, 1] . Can anything be said about the existence of Give reasons for your answer.

Find the limits in Exercises 11–22. \(\lim _{x \rightarrow-7}(2 x+5)\) Equation Transcription: Text Transcription: Lim x right arrow-7 (2x+5)

\(\text { If } f(1)=5, \text { must } \lim _{x \rightarrow 1} f(x) \text { exist? If it does, then must } \lim _{x \rightarrow 1} f(x)=5 ? \text { Can we conclude anything }\) about \(\lim _{x \rightarrow 1} f(x) ?\)? Explain. Equation Transcription: Text Transcription: f(1)=5 Lim x right arrow 1 f(x) Lim x right arrow 1 f(x)=5 Lim x right arrow 1 f(x)

\(\text { If } \lim _{x \rightarrow 1} \mathrm{f}(x)=5, \text { must } \mathrm{f} \text { be defined at } \mathrm{x}=1 \text { ? f it is, must } \mathrm{f}(1)=5 \text { ? }\) Can we conclude anythingabout the values of \(f \text { at } x=1 ?\) Explain. Equation Transcription: Text Transcription: Lim x right arrow 1 f(x) =5 x=1

Problem 12E Find the limits in Exercises 11–22.

Problem 13E Find the limits in Exercises 11–22.

Find the limits in Exercises 11–22. \(\lim _{x \rightarrow 2} \frac{x+3}{x+6}\) Equation Transcription: Text Transcription: Lim x right arrow 2 x+3/x+6

Problem 14E Find the limits in Exercises 11–22.

Find the limits in Exercises 11–22. \(\lim _{x \rightarrow-1} 3(2 x-1)^{2}\) Equation Transcription: Text Transcription: Lim x right arrow -1 3(2x-1)^2

Find the limits in Exercises 11–22. \(\lim _{s \rightarrow 2 / 3} 3 s(2 s-1)\) Equation Transcription: Text Transcription: Lim s right arrow ? 3s(2s-1)

Find the limits in Exercises 11–22. \(\lim \limits_{y\rightarrow2}\ \frac{y+2}{y^2+5y+6}\) Equation Transcription: Text Transcription: Lim y right arrow 2 y+2/y^2+5y+6

Find the limits in Exercises 11–22 \(\lim _{z \rightarrow 0}(2 z-8)^{1 / 3}\) Equation Transcription: Text Transcription: Lim_z right arrow 0 (2z-8)^1/3

Problem 22E Find the limits in Exercises 11–22.

Problem 19E Find the limits in Exercises 11–22.

Find the limits in Exercises 11–22. \(\lim _{h \rightarrow 0} \frac{3}{\sqrt{3 h+1}+1}\) IEquation Transcription: Text Transcription: Lim_h right arrow 0 3/sqrt 3h + 1 + 1

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow-3} \frac{x+3}{x^{2}+4 x+3}\) Equation Transcription: Text Transcription: Lim_x right arrow 3 x + 3/x^2 + 4x + 3

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25}\) Equation Transcription: Text Transcription: Lim_x right arrow 5 x - 5/x^2-25

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow-5} \frac{x^{2}+3 x-10}{x+5}\) Equation Transcription: Text Transcription: Lim_x right arrow -5 x^2 + 3x - 10/x + 5

Problem 26E Limits of quotients Find the limits in Exercises 23–42.

Problem 27E Limits of quotients Find the limits in Exercises 23–42.

Find the limits in Exercises 23–42. \(\lim _{t \rightarrow-1} \frac{t^{2}+3 t+2}{t^{2}-t-2}\) Equation Transcription: Text Transcription: Lim_t right arrow -1 t^2 + 3t + 2/t^2 - t - 2

Problem 29E Limits of quotients Find the limits in Exercises 23–42.

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow 1} \frac{\frac{1}{x}-1}{x-1}\) Equation Transcription: Text Transcription: Lim_x right arrow 1 1/x - 1/x - 1

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow 0} \frac{\frac{1}{x-1}+\frac{1}{x+1}}{x}\) Equation Transcription: Text Transcription: Lim_x right arrow 0 1/x - 1 + 1/x + 1 / x

Find the limits in Exercises 23–42. \(\lim _{u \rightarrow 1} \frac{u^{4}-1}{u^{3}-1}\) Equation Transcription: Text Transcription: Lim_u right arrow 1 u^4 - 1/u^3 - 1

Problem 30E Limits of quotients Find the limits in Exercises 23–42.

Problem 34E Limits of quotients Find the limits in Exercises 23–42.

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}\) Equation Transcription: Text Transcription: Lim_x right arrow 9 sqrt x - 3/x - 9

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow 4} \frac{4 x-x^{2}}{2-\sqrt{x}}\) Equation Transcription: Text Transcription: Lim_x right arrow 4 4x - x^2/2 - sqrt x

Problem 37E Limits of quotients Find the limits in Exercises 23–42.

Problem 39E Limits of quotients Find the limits in Exercises 23–42.

Problem 40E Limits of quotients Find the limits in Exercises 23–42.

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow-3} \frac{2-\sqrt{x^{2}-5}}{x+3}\) Equation Transcription: Text Transcription: Lim_x right arrow -3 2 - sqrt x^2 - 5/x + 3

Find the limits in Exercises 23–42. \(\lim _{x \rightarrow-1} \frac{\sqrt{x^{2}+8}-3}{x+1}\) Equation Transcription: Text Transcription: Lim_x right arrow -1 sqrt x^2 + 8 - 3/x + 1

Problem 42E Limits of quotients Find the limits in Exercises 23–42.

Find the limits in Exercises 43–50. \(\lim _{x \rightarrow 0}(2 \sin x-1)\) Equation Transcription: Text Transcription: Lim_x right arrow 0(2 sin x - 1)

Find the limits in Exercises 43–50.

Find the limits in Exercises 43–50.

Find the limits in Exercises 43–50. \(\lim _{x \rightarrow \pi / 3} \tan x\) Equation Transcription: Text Transcription: Lim_x right arrow pi/3 tan x

Problem 48E Limits with trigonometric functions Find the limits in Exercises 43–50.

Find the limits in Exercises 43–50. \(\lim _{x \rightarrow 0} \frac{1+x+\sin x}{3 \cos x}\) Equation Transcription: Text Transcription: Lim_x right arrow 0 1 + x + sin x/3 cos x

Problem 49E Limits with trigonometric functions Find the limits in Exercises 43–50.

Find the limits in Exercises 43–50. \(\lim _{x \rightarrow 0} \sqrt{7+\sec ^{2} x}\) Equation Transcription: Text Transcription: Lim_x right arrow 0 sqrt7 + sec^2 x

Suppose \(\lim _{x \rightarrow 0} f(x)=1\) and \(\lim _{x \rightarrow 0} g(x)=-5\) Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and(c) of the following calculation. \(\lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{(f(x)+7)^{2 / 3}}=\) \(\frac{\lim _{x \rightarrow 0}(2 f(x)-g(x)}{\lim _{x \rightarrow 0}(f(x)+7)^{2 / 3}}\) \(=\frac{\lim _{x \rightarrow 0}\left(2 f(x)-\lim _{x \rightarrow 0} g(x)\right.}{\left(\lim _{x \rightarrow 0}(f(x)+7)\right)^{2 / 3}}\) \(=\frac{\lim _{x \rightarrow 0}\left(2 f(x)-\lim _{x \rightarrow 0} g(x)\right.}{\left(\lim _{x \rightarrow 0} f(x)+\lim _{x \rightarrow 0} 7\right)^{2 / 3}}\) \(=\frac{(2)(1)-(-5)}{(1+7)^{2 / 3}}=\frac{7}{4}\) Equation Transcription: Text Transcription: lim_x right arrow0 f(x) = 1 lim_x right arrow0 g(x) = -5 Lim_x right arrow0 2f(x) - g(x)/(f(x) + 7)^2/3= lim_x right arrow 0 (2f(x) - g(x)/ lim_x right arrow 0 (f(x) + 7)^2/3 =Lim_x right arrow0 (2f(x) - Lim_x right arrow0 g(x)/(Lim_x right arrow0 (f(x) + 7))^2/3 =Lim_x right arrow0 (2f(x) - Lim_x right arrow0 g(x)/(Lim_x right arrow0 f(x) + x07)^2/3 =(2)(1) - (-5)/(1 + 7)^2/3=7/4

Problem 52E Let and Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

Suppose \(\lim _{x \rightarrow c} f(x)=5\) and \(\lim _{x \rightarrow c} g(x)=-2\). Find a. \(\lim _{x \rightarrow c} f(x) g(x)\) b. \(\lim _{x \rightarrow c} 2 f(x) g(x)\) c. \(\lim _{x \rightarrow c}(f(x)+3 g(x))\) D. \(\lim _{x \rightarrow c} \frac{f(x)}{f(x)-g(x)}\) Equation Transcription: Text Transcription: Lim_x right arrow c f(x)=5 Lim_x right arrow c g(x)=-2 Lim_x right arrow c f(x) g(x) Lim_x right arrow c f(x)+ 3g(x)) Lim_x right arrow c f(x)/ f(x)-g(x)

Problem 54E Suppose and Find

Suppose that \(\lim _{x \rightarrow 2} p(x)=4, \lim _{x \rightarrow 2} r(x)=0\) and \(\lim _{x \rightarrow 2} s(x)=-3\). Find (a) \(\lim _{x \rightarrow 2}(p(x)+r(x)+s(x))\) (b) \(\lim _{x \rightarrow 2}(p(x)+r(x)+s(x))\) (c) \(\lim _{x \rightarrow 2}(-4 p(x)+5 r(x)) / s(x)\) Equation Transcription: Text Transcription: lim_x right arrow 2p(x)=4, lim_x right arrow 2 r(x)=0 lim_x right arrow 2 s(x)=-3 lim_x right arrow 2(p(x)+r(x)+s(x)) lim_x right arrow 2 p(x) . r(x) . s(x) lim_x right arrow 2 (-4p(x) + 5r(x))/s(x)

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of \(x\) and function \(f\). \(f(x)=x^{2}, \quad x=1\) Equation Transcription: Text Transcription: lim_h right arrow 0 f(x+h) - f(x)/h x f f(x) = x^2, x = 1

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of \(x\) and function \(f\). \(f(x)=x^{2}, \quad x=-2\) Equation Transcription: Text Transcription: lim_h right arrow 0 f(x+h) - f(x)/h x f f(x) = x^2, x = -2

Problem 59E Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function ƒ. ƒ(x) = 3x - 4, x = 2

Problem 55E Suppose and Find

Problem 60E Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function ƒ. ƒ(x) = 1/x, x = -2

Problem 61E Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function ƒ.

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of \(x\) and function \(f\). \(f(x)=\sqrt{3 x+1}, \quad x=0\) Equation Transcription: Text Transcription: lim_h right arrow 0 f(x+h) - f(x)/h x f f(x) = sqrt 3x + 1, x = 0

Problem 63E If for -1 ? x ? 1, find

If \(2-x^{2} \leq g(x) \leq 2\) cos x for all \(x\), find \(\lim _{x \rightarrow 0} g(x)\). Equation Transcription: Text Transcription: 2 - x^2 ? g(x) ? 2 cos x x lim_x right arrow 0 g(x).

a. Suppose that the inequalities \(\frac{1}{2}-\frac{x^{2}}{24}<\frac{1-\cos x}{x^{2}}<\frac{1}{2}\) hold for values of \(x\) close to zero. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}\)? Give reasons for your answer. b. Graph the equations \(y=(1 / 2)-\left(x^{2} / 24\right), y=(1-\cos x) / x^{2}\), and \(y=1 / 2\) together for \(-2 \leq x \leq 2\). Comment on the behavior of the graphs as \(x \rightarrow 0\). Equation Transcription: Text Transcription: 1/2-x^2/24 <1 - cos x/x^2<1/2 x about lim_x right arrow 0 1 - cos x/x^2 y = (½) - (x^2/24), y = (1 - cos x)/x^2 y = ½ -2 ? x ? 2 x right arrrow 0

Problem 65E a. It can be shown that the inequalities hold for all values of x close to zero. What, if anything, does this tell you about Give reasons for your answer. b. Graph y = 1 – (x2/6), y = (x sin x)/(2 - 2 cos x), and y = 1 together for -2 ? x ? 2. Comment on the behavior of the graphs as

Problem 67E You will find a graphing calculator useful for Exercises 67–76. Let ƒ(x) = (x2 – 9)/(x + 3). a. Make a table of the values of ƒ at the points x = -3.1, -3.01, -3.001, and so on as far as your calculator can go. Then estimate What estimate do you arrive at if you evaluate ƒ at x = -2.9, -2.99, -2.999, … instead? b. Support your conclusions in part (a) by graphing ƒ near c = -3 and using Zoom and Trace to estimate y-values on the graph as c. Find algebraically, as in Example 7. EXAMPLE 7 Evaluate Solution We cannot substitute x = 1 because it makes the denominator zero. We test the numerator to see if it, too, is zero at x = 1. It is, so it has a factor of (x – 1) in common with the denominator. Canceling the (x – 1)’s gives a simpler fraction with the same values as the original for x ? 1: Using the simpler fraction, we find the limit of these values as by substitution: See Figure 2.11.

Problem 69E Let G(x) = (x + 6)/(x2 + 4x – 12). a. Make a table of the values of G at x = -5.9, -5.99, -5.999, and so on. Then estimate What estimate do you arrive at if you evaluate G at x = -6.1, -6.01, -6.001, … instead? b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as c. Find algebraically.

Problem 70E Let h(x) = (x2 - 2x – 3)/(x2 - 4x + 3). a. Make a table of the values of h at x = 2.9, 2.99, 2.999, and so on. Then estimate What estimate do you arrive at if you evaluate h at x = 3.1, 3.01, 3.001, … instead? b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as c. Find algebraically.

You will find a graphing calculator useful for Exercises 67–76. Let \(F(x)=\left(x^{2}+3 x+2\right) /(2-|x|)\). a. Make tables of values of \(F\) at values of \(x\) that approach \(c = -2\) from above and below. Then estimate \(\lim _{x \rightarrow 2} F(x)\). b. Support your conclusion in part (a) by graphing F near \(c = -2\) and using Zoom and Trace to estimate \(y\)-values on the graph as \(x \rightarrow-2\). c. Find \(\lim _{x \rightarrow 2} F(x)\) algebraically. Equation Transcription: Text Transcription: F(x) = (x2 + 3x + 2)/(2 - |x|) F x c = -2 lim_x right arrow 2F(x) y x ? -2 lim_x right arrow 2F(x)

Problem 68E Let a. Make a table of the values of g at the points x = 1.4, 1.41, 1.414, and so on through successive decimal approximations of Estimate b. Support your conclusion in part (a) by graphing g near and using Zoom and Trace to estimate y-values on the graph as c. Find algebraically.

Problem 74E Let G(t) = (1 - cos t)/t2. a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate b. Support your conclusion in part (a) by graphing G near t0 = 0.

Problem 73E Let g(?) = (sin ?)/?. a. Make a table of the values of g at values of ? that approach ?0 = 0 from above and below. Then estimate b. Support your conclusion in part (a) by graphing g near ?0 = 0.

Problem 75E Let ƒ(x) = x1/(1-x) . a. Make tables of values of ƒ at values of x that approach c = 1 from above and below. Does ƒ appear to have a limit as If so, what is it? If not, why not? b. Support your conclusions in part (a) by graphing ƒ near c = 1.

Problem 76E Let ƒ(x) = (3x – 1)/x. a. Make tables of values of ƒ at values of x that approach c = 0 from above and below. Does ƒ appear to have a limit as If so, what is it? If not, why not? b. Support your conclusions in part (a) by graphing ƒ near c = 0

If x4 ? f(x) ? x2 for x in [-1, 1] and x2 ? f(x) ? x4 for x < -1 and x > 1, at what points c do you automatically know ? What can you say about the value of the limit at these points?

Problem 78E Suppose that g(x) ? f(x) ? h(x) for all x ? 2 and suppose that Can we conclude anything about the values of ƒ, g, and h at x = 2? Could f(2) = 0? Could Give reasons for your answers.

Problem 79E If find

If \(\lim _{x \rightarrow 2} \frac{f(x)}{x^{2}}=1\), find \(\lim _{x \rightarrow-2} f(x)\) \(\lim _{x \rightarrow-2} \frac{f(x)}{x}\) Equation Transcription: Text Transcription: lim_ x right arrow 2 f(x)/x^2=1 lim_ x right arrow -2f(x) lim_ x right arrow 2 x-2f(x)/x

(a) If \(\lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=3\), find \(\lim _{x \rightarrow 2} f(x)\). (b) if \(\lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=4\), find \(\lim _{x \rightarrow 2} f(x)\) Equation Transcription: Text Transcription: lim_ x right arrow 2 f(x) - 5/x - 2=3 lim_ x right arrow 2f(x) lim_ x right arrow 2 f(x) - 5/x - 2=4 lim_ x right arrow 2f(x)

Problem 82E If , find

Problem 83E a. Graph g(x) = x sin (1/x) to estimate zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a proof.

1. Graph \(h(x)=x^{2} \cos \left(1 / x^{3}\right)\) to estimate \(\lim _{x \rightarrow 0} h(x)\), zooming in on the origin as necessary. 2. Confirm your estimate in part (a) with a roof. Equation Transcription: Text Transcription: h(x) =x2cos (1/x3) lim_x right arrow 0 h(x)

Problem 86CE In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit.

In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit. \(\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}\) Equation Transcription: Text Transcription: lim_x right arrow 2 x^4 - 16/ x - 2

In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit. \(\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x-1}}{x}\) Equation Transcription: Text Transcription: lim_x right arrow 0 3 sqrt 1 + x - 1/ x

In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit. \(\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x^{2}+7-4}}\) Equation Transcription: Text Transcription: lim_x right arrow 3 x^2 - 9/ sqrt x^2 + 7 - 4

In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit. \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x \sin x}\) Equation Transcription: Text Transcription: lim_x right arrow 0 1 - cos x/ x sin x