12. Suppose f € C[0, 1]. Prove that lim | f(a" )da = f(0). n>00

Q: II. Prove that for each of the following lim (x,y)→(0,0) f(x, y) does not exist. ,2,,2 4. f(x,y) x4…

A: limx, y→0, 0x2y2x4+y4

Q: Theory and Examples

A:

Q: 4] Compute the following limits, using the Algebraic Limit Theorem,where (x„) → 0. 1+2x, a) lim…

A: a) lim 1+2xn1+2xn-3x22 Since limit of xn is zero . Using lim cxn=climxn So, lim 2xn=2limxn=20=0…

Q: Estimate the given limit numerically, if it exists. lim, 0 (A) O 1 O does not exist

A: estimate the given limit, numerically, limx→0x2x+1

Q: in z2 Q5. Find the value of the limit: lim in z²+1 .- 기 z→e 2

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 1. Use the e - ô definition of the limit to show the following: (a) lim(5x – 2) = 3 (b) lim VT = 0

A:

Q: 2. Compute the following limits at infinity and give reasons for your answers. 1 a) lim b) lim I00…

A:

Q: 4 cos(20ax) – 4 Let a + 0. Compute the limit lim 25a2 x2

A:

Q: 1) Show that the limit doesn't exists : ysinx 3) lim (x,y)→(0,0) x²+y2

A: To show that the limit of a multivariable function does not exist,…

Q: In (x +h) – In (x) is When x = 2e, lim h→0 h

A: Substitute x=  2e above

Q: Compute the exact value of the given limit. V5x2 + 1 lim 6x

A:

Q: evaluate the limit. lim e2 'i+ In(t + 1)j + 4k

A:

Q: 1 Discuss the existence of the limit lim x 0 x2"

A:

Q: Evaluate the following limits (if exist): 1-cos(ax) (c) lim- a+0 (d) lim/Vxr'+x-Vx°-cx), where c is…

A: To evaluate the given limit we are using L- Hospital rule

Q: 2. Use L'Hospital's rule to evaluate the following limit. cosh(3x) – 1 lim x→0 x2

A:

Q: Solve the following limit by definition (ii) (4 – 2x, x 1 lim f(x) = 2 if f(x) = %3D x→1

A:

Q: What is the limit as x goes to 0 for (e^x-1-x)/x^2?

A: Please refer the attached image for complete solution.

Q: in Q5. Find the value of the limit: lim in — е 2 2+1 Z→e 2

A:

Q: Suppose that a, > 0 and converges. lim na, = 0. Prove that Ea, n 00

A:

Q: Find the limit. lim x-2/In x X x-0+ O -2 O e?

A:

Q: Show that the following limit does not exist. ху lim (x. y)-(0, 0) 3x2 +2y2

A:

Q: Find the required limit or indicate that it does not exist. lim o+ (in (t6), 12 in t O 6i O Does not…

A: We have to find out

Q: Suppose that f is decreasing on [a,b]. Prove that that b. lim Rpf) exists and equals to Ilp||+0

A:

Q: 2) If x3 s f(x) sx for x in [-1,1), find lim f(x) if it exists. x-0 A) 0 B) Does not exist C) -1 D)…

A:

Q: min 1) In formally proving that lim (x + x) 42, let e > 0 be arbitrary. Choose & %3D Determine m. m…

A:

Q: Verify numerically that:

A: To verify: limh→0eh-1h=1

Q: 4. Evaluate the limit of as e approaches 0. 2a A 2 B. % CO D. infinity

A:

Q: 2 Show that the following limit does not exist. lim Ty Cos y (r.9)-(0.0) 3r+ y²

A:

Q: 2. Let r > 1. Prove that lim r" = 00,

A:

Q: Use the Squeeze Theorem to prove lim = 0. x00 x+ e- For x 0, since 0 < e* <1, we have

A:

Q: Compute the following limit, or verify that it does not exist: xY lim (x,y)→(0,0) 2x² + 3y2

A: Given that limx,y→0,0xy2x2+3y2 fx,y=xy2x2+3y2 (i) Put x=0 and then y=0 in fx,y find the value of f1…

Q: Prove, using the formal definition, that lim→-1 (T+1)² +0o. %3D

A: We have to find the limit limx→-13x+12=+∞.

Q: lim el -1 dt 61-12-e

A:

Q: Find the maximum value of & > 0 that satisfies the limit claim corresponding to e = 0.3, that is,…

A: In the question it is asked to calculate the maximum value of δ for the given function limx→59-x5=8

Q: 10. Prove that the statement "If lim f(r) exists, then f is continuous at a" is false by using a…

A: See the solution below.

Q: Use the definition of the limit (e – 8) to show that 1 lim - = i. 2+-i z

A:

Q: Use two different methods to find the limit where m &gt; 0, n &gt; 0, and x &gt; 0

A: The given limit is: limx→∞lnxmlnxn where, m &gt; 0, n &gt; 0, and x &gt; 0 First method: To evaluate…

Q: Prove that ƒ is continuous at c if and only if lim h--&gt;0 ƒ(c + h) = ƒ(c).

A: If part:

Q: 1. Show that the limit lim (2,y)(0,0) In(1 – a2)+ y* x² + y4 does not exist.

A:

Q: Compute the following limits, using the Algebraic Limit Theorem,where (xn) → 0. 1+2n a) lim…

A: Consider the given information. (a) Consider the given expression. limxn→01+2xn1+2xn−3x2n Apply the…

Q: Suppose that f is decreasing on [a,b] Prove that that lim Rpf) exists and equals to

A: let ∈&gt;0 be any real number  let P={a=t0&lt;t1&lt;t2&lt;......&lt;tn=b} be a partition of [a,b]…

Q: Compute the limit 1 e¯nt lim dt. VE n-00

A: Given: limn→∞∫01e-nttdt

Q: The limit fx= X2 XスT 2メ-ズ Cos X ,X < TI at Will be 1-itexisted and eanals T 2-It existed and equals…

A: In the question we have to find correct option.

Q: 1. Prove the following limits x 4y4 G) lim (x,y) (0,0) (x²+y4)3 = Does not exist

A: Solution is given below:

Q: In (r+1) lim- t

A: The given limit is: limt→∞lnt2+1t When substituting the direct limit, it becomes…

Q: Estimate the given limit numerically, if it exists lim, 1* (E) O does not exist O 4

A: Explanation of the answer is as follows

Q: 2. Use e -d of the limit to prove that lim z= 4

A: Given below the detailed solution

Q: Show that the following limit does not exist: (x – y²)2 lim (r,y) (0,0) ² + y4

A:

Q: 27 Let f(z) = エー3 Find lim f(r). Enter I for oo, -I for - 00, and DNE if the limit does not exist.…

A: In this question we have to find the limit of the given function.