Fact 1: | arctan t<14 for any real number t. Fact 2: t <14 for any tE (-1, 1]. (a) Prove by using the triangle inequality that 2 arctan(2r) + ar|< (a +4)|r| if |r|S1. must indicate where you are using the triangle inequality in your solution. (b) Prove by using the definition of a limit that lim (2 arctan(2r) + ar) = 0.

Q: Find the limit using the squeeze theorem.   Assume the function g satifies the inequality…

A:

Q: Prove the limit statements lim x sin = 0 y=x sin 2

A: Prove the limit statements     limx→0x sin1x=0

Q: 2n3 +1 1 Prove, using definition of limit, that lim 4n3 – n 2 n-00 -

A:

Q: Suppose that a function f(x) is defined for all real values of x in [-1,1]. Can anything be said…

A:

Q: 3 Find the limit: lim t 3 et – 1 - (Enter undefined if the limit does not exist.)

A:

Q: 1. Evaluate the limit if it exists. If it does not exist, justify. 2 cos(3x? + 5) + [6x² +2] (a) lim…

A: Please see the below picture for detailed solution.

Q: 2. Use the ɛ-d definition of the limit of a function to prove that lim = 3. I-»3 T – 2

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

Q: (z+2 z< 0 sin (x) 0<æ Let F (x) = Then lim F (x) =

A:

Q: 1. Evaluate each of the following limits. You may use any method you like, but you must evaluate at…

A: Apple the L, Hospital rule

Q: Question 10 Find the limit, if it exists Limy-9 (x+9) / (1/x+ 1/9)

A: The given limit is limx→9x+91x+19.

Q: QUESTION 11 Compute the limit. Justify your answer. 7/2-19x/2+7 3/2+7 lim 3x-2x +11

A: We need to find the value of limx→∞ 10x72-19x32+73x4-2x+11

Q: Find the limit (1+2)(1+2x)(1+3z)-1 lim 0 O 3 6. O 1 None of these

A: We need to find the limit limx→01+x1+2x1+3x-1x.

Q: The limit as x approaches infinty of (x+3^x)/(x^2-3^x) Evaulate using L'Hospital's Rule providing a…

A:

Q: Find the limit. lim X-2 x+2 O 1/2 -00 O Does not exist

A: Answer is limit does not exist

Q: Estimate the limit numerically or state that the limit doesn't exist lim h→0 cos(9/h)

A: We will pick some values for x. Then we will find values of y for those xs.

Q: 1. Solve the following limits. (a) lim 1-0 V1+ x² x In x (b) lim 1400 x + In x 2+ sec (3x) (c) lim…

A:

Q: 1. Find the limit: lim 1-e-2t 1t, ,1+t² tan t00 '1-t2

A: We need  to find limit.

Q: 2x sin-x lim х—0 еx — Х — 1

A:

Q: Find the limits lim x→ -1+ cos-1x

A: We find the limit by substitution

Q: f (х, у) %3D х4 + 4y2/x2 + 2y2 f function is defined as. Does f function have a limit at point…

A: We will evaluate the limits using two paths and see if they are same. If not same, the limit does…

Q: What are the limits of both parts of this question?

A: In part a, there is no bound, so the answer of the limit is infinity.

Q: 1 - cos h Investigate lim h-0 numerically or graphically. Then h2 prove that the limit is equal to…

A: The limit to evaluate is  limh→01-coshh2 To find the limit numerically calculate the values of…

Q: 2+4x 7. Prove lim = 2 using the e-ô definition of the limit. 3 %3D X-1

A: Hey, since there are multiple questions posted, we will answer first question of the first image. If…

Q: 1. Find the limit (if it exists). (If an answer does not exist, enter DNE.) 2 - x im x² – 4 sin 2x…

A: Find the limit of the given function  1) (2-x)/(x2-4) , when x tends to 2 .  2) sin(2x)/sin(3x) ,…

Q: Find the limit, If it exists. 8 lim 9 cos 5x

A:

Q: QUESTION 4 Compute the limit below. Justify your answer. (5x +8h)2-25x2 lim- h

A:

Q: Find the limit. 6x + 2 lim e 6x+ 2 =(Type an exact answer in terms of e.)

A:

Q: (7) ifa lim has a point that the equalio- prove PCz,y) is above L and only if 3x +2y >4

A:

Q: QUESTION 2 Prove using the definition of a limit that lim x² + y2 -y = 4. >

A:

Q: compure the limit ) um X-7-3 X3-9x 6+2X Sin/2x) 2) lim sini2x) X-7ut x2 3 X 3) lim X-7-00 イ4メ2tメt5

A: Given: -

Q: Q4: - Evaluate the limit of the following: x3–1 sin 2x 1-V 1) lim 2) lim x→0 2x²+x 3) lim x→1 1-x…

A: When we get 00form, we can us  L' Hospitals Rule to calculate the limits. In this rule μx→a…

Q: 7. Compute the limit to infinity: 4-2+3r+1 a lim 7+3r-r+ 12

A:

Q: 5. Evaluate the limits using any of the methods you know. Show your work. In(1+4x) c. lim tan 2x

A: find the limit

Q: 5. Use the concept of limits to solve for the following problem iz3 – 1 lim z-1 z + i

A:

Q: 4. Use the e - 6 definition of limit to show that lim (2: +3i) = 2 + 5i. 2+1+1

A:

Q: 4. a) Explain using theorems from Section 2.5 why the function is continuous at every number in its…

A:

Q: cos 0-1 lim 00 40 4.

A:

Q: 2. Find the limit using Squeeze theorem 1 lim-2|x – 2| sin( r - 2

A: Squeeze Theorem- Let f, g and h be functions defined on an interval I. Let c∈I be its limit point.…

Q: Use the Squeeze Theorem to show that lim (x² cos 21xx) = 0. Illustrate by graphing the functions…

A:

Q: sin²(2x) lim 1-sin² x 2 lim (2x – 1)vx + 4

A: To find the given limits.

Q: 1c. Use L'Hopital's Rule to find the Limit: lim          (1 - x)tan(pix/2) x-&gt;1-

A: We will use tan z = 1/cot z formula

Q: 4 Using the e – 8 definition of limit, prove that lim x→3 3x 1 1 2

A: To prove the limit of given function using the epsilon delta definition of limit which states that…

Q: 1. Find the following limits if they exist, whether they are a real number or too. If a limit does…

A:

Q: 6 -- if a - 3 Calculate the following limits. Enter "DNE" if the limit does not exist. lim f(x) =…

A:

Q: 1 11 lim X- 1 x2 + 9x – 10 Give your answer in exact form. If the limit does not exist, enter DNE.

A:

Q: Find the limit. 1 lim COS x-4 4

A:

Q: Can anyone solve this question please ? Many Thanks

A:

Q: Compute the following limit.“ KC'arctan(2t³ – 1) dt] du (x – 1) In r lim Be sure to explicitly…

A: We have to evaluate a limit and we will use the L'hospital's Rule to find the answer.