31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) – sin a = h h/2 | sin x Then use the inequality < 1 for x # 0 to show that |sin(a + h) – sin a| < |h| for all a. Finally, prove rigorously that lim sin x = sin a.

Q: Evaluate using the Squeeze Theorem

A: To Evaluate: limx→0xsin1x2 Concept: Squeeze Theorem:limx→afx≤limx→agx≤limx→ahxand if…

Q: sin 7x lim x→0 sin 3x

A: The given limit is limx→0sin7xsin3x

Q: b. Evaluate the following limits: 7x cos)-7x lim- [sin(-2x)sin(5x)sin(7x)] lim 5x2 2x cos(x) X-0

A: We write the given limit in the following format

Q: evaluate using Theorem 2 as necessary.

A:

Q: Evaluate the following limits: (a) limx→0 4x/sin 5x (b) limx→π sin( π/2)/5x (c) limx→0…

A: a) limx→04xsin(5x)=45 b) limx→πsinπ25x =sinπ2limx→π15x                    = 015π limx→πsinπ25x = 0…

Q: Suppose the inequality (x² – 4x + 3) < f (x) < cos ( TX *) holds for all values of x. 2 Find lim f(…

A:

Q: 1- cos X lim x→0 1- Prove that

A: As per Bartleby's expert answering policy, we cannot answer more than one question so kindly post…

Q: Use the relation lim = 1 to determine the limit. 0→0 1- cos 0 lim sin 170 0→0

A:

Q: - Evaluate only thre tan 30 lim 0 sin 80 x+1 lim X00x2+3 ii. lim,sx.g(x) if lim,s = 2 *+g(x)…

A: Solve only three subparts

Q: b) If x sin x 2 - 2 cos x hold for all values of x close to zero. Find limx0 xsinx ? 2-2cosx

A: Given, limx→0xsinx2-2cosx

Q: Determine which of the following is a correct step to help show lim x*sin = 0 by using the Squeeze…

A:

Q: Hence, determine whether the function is continuous at x = 2x. ) Find the value of k such that the 5…

A:

Q: Evaluate the following limits tan x – sin x sin(x – 2) (a) lim x→0 x(-1+ cos³ 2x) (b) lim x→2 Vx + 3…

A:

Q: We can use direct substitution in computing limits provided there is nodivision by zero, which is…

A:

Q: The value of limit x→0 of {sin (a + x) – sin (a – x)}/x is

A: we have to find the value of  limx→0sin(a+x)-sin(a-x)x

Q: Find the value of each limit: 1- cos x а. lim x→0 x+x b. lim sin(x+ h) – sin(x) h h→0

A:

Q: Compute the following limits if they exist. sec x – 1 (a) lim I0- x – sin a (b) lim I40 x – tan x 2…

A: "Since you have posted a question with multiple sub- parts, we will solve first three sub -parts…

Q: Compute the following limits if they exist. tan x x – sin x (а) lim 0+ Vx (b) lim (c) lim tan x- seC…

A: We have to find the limit

Q: Evaluate the following limits: 1. 2 sin x - sin(2x) -x-sin x (a) lim 25 -t2 (b) lim 15 tl 11m,…

A: Hi, since there are multiple sub-parts in the questions posted, we will answer first three subparts.…

Q: I. Evaluate the following limits. 1 1 1. lim x – 1 et — е x→1+ - - tan x 2. lim (sin x x→0+

A:

Q: B. Evaluate only three of the following limits tan 30 lim 0-0 sin 80 x+1 i. ii. lim + zX co-x 1 = 2…

A:

Q: evaluate using the Squeeze Theorem.

A: We know that sin is always  between -1 to 1 Multiply all sides by (x-1) and then take limit x-&gt;…

Q: Suppose f(x,y) = sin-' (6 – 2x2 – y²) and g(x,y) = 4x2 – y2. Determine the continuity of the…

A:

Q: 14. Prove that sin x cos x from dx first principles and be sure to prove all limits used.

A: Proof of limit which is used to prove derivative of sin(x) is cos(x).

Q: Evaluate the following limits: cos()- lim x-0 )-7x lim [sin(-2x)sin(5x)sin(7x)] 5x2 2x°cos(x)

A:

Q: Evaluate the following limits: 1 1 (i) lim{ sin -+cos- X X 1/x 1/x +a, 1/x a1 + (ii) lim ..... ,a1…

A:

Q: 2. Evaluate the following limit. 1/x lim (1– sin 3x)"* x→0*

A: Given, limx→0+1-sin3x1x    We have to evaluate this limit

Q: In evaluating 2 dx 5₂₁²= is same as evaluating lim_cos-¹(f(x))|c. Determine the value of f(x) if…

A: Derivative of cos(x) = - sin(x) We have to find the value of f(x) at x=0.96

Q: lim sin?(2 – x) cot²(x² – x – 2) - X→2

A: We have to evaluate the limit at x= 2.

Q: (a) Evaluate two of the following limits: 1-cos ax sin( 1-x² ) tan 2x (i) lim x→0 1-cos bx (ii) lim…

A:

Q: Compute the following limits if they exist. sec x – 1 (a) lim x – sin x (b) lim x→0 x – tan x 2 - VI…

A:

Q: 3. Use Squeeze Theorem to evaluate lim 1-0+ sin(In r) In(sin æ)"

A:

Q: 2. Calculate the following limits: (a) COS x – 1 lim x40 sin x In(1 + x)

A: To find the limit Lim(x tends to 0) [cosx-1]/sinxln(1+x)

Q: )Use the Sandwich Theorem to show that lim cos x-1 =0. x2 Find f'(x) for f(x) = sec(vx2 + 1).

A:

Q: 30. Use the fact that lim x→0 sin x X = 1 (shown in video) to show lim x-0 1 - cos x X = 0

A: we will find the required.

Q: . Evaluate the following limits. 1.lim cos .2 () 2. lim sin x→0 (- 3. lim 3¬× X→-2

A: 1. limx→πcosx22. limx→0sinxx3. limx→-23-x

Q: Evaluate the following limits. 2 sin 5x a. lim x→0 3x b. lim sin 5x cot 3x x→0 d. lim (sec x – tan…

A: Hey, since there are multiple questions posted, we will answer first three question. If you want any…

Q: エ→2ー Prove that lim sin(1/x) does not exist. Lot II

A:

Q: (1point Use continuity to evaluate lim sin(-8x + sin 5x) Enter I for oo, -I for -co, and DNE if the…

A:

Q: Compute the following limits if they exist. sec x – 1 (а) lim x – sin x (b) lim I-0 x – tan r

A:

Q: Use continuity to evaluate lim sin(2x + sin 3x) Enter I for o, -I for –o, and DNE if the limit does…

A:

Q: I. Evaluate the following limits. 1 1 1. lim x – 1 et — е x→1+ - - 2. lim (sin x)tan a x→0+

A:

Q: 1 lim x→0 x sin x x2 О 14 O 1/7 О 16 O 1/5

A: To find limx → 0 1x sin x - 1x2

Q: Express the limit lim (4 sin (2rx:) + 6)Ar; over [2, 3] as an integral. n- 00 i=1 Provide a, b and…

A:

Q: Calculate the following limits. cos x 1) lim (tan x)cos* x² – sin?2x 2) lim x→0 x2

A:

Q: Evaluate using Theorem 2 as necessary.

A: Given: limx→0 x2sin2(x)

Q: Q.3/ Find the limits for the following: 1) lim sin 7x 2) lim x0 sin 5x 3) lim V-3 x→9 ズ-9 x1x3-

A: We will find the limit.

Q: Express the limit lim > (5 sin(27x; ) + 8)Ax; over [1, 5] as an integral. n→ 00 i=1 Provide a, b and…

A: To find the solution

Q: Use e-ð to prove lim,1 sin x = sin 1. [Hint: we know sin(1 +h) = sin(1) cos(h) + cos(1) sin(h),…

A: We have to prove: limx→1sin(x)=sin1