lim (1,4) (0,0) x2 + y2 Prove that

Q: lim (4x – 3y)= -11 Prove that (r.,v(-2,1)"

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Q: lim a'/z 2-

A: Given:

Q: 1 (ii) lim z-»b b - x

A: The given limit is limx→b-1b-x

Q: Find the limits

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Q: lim (x,y)→(0,0) In(2 + x² + y?) x² + y?

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Q: lim x-4y (x,y)- (0,0) 6y+7x .A does not exist.

A: To calculate the given limit: limx,y→0,0x−4y6y+7x

Q: 2 i (z-i) lim z?+1 1 -1 о 2i

A: 1 Just use L'Hospitals rule.

Q: z² + iz lim z3 + 1 z- ob ||

A: We are given : limz→∞z2+izz3+1=limz→∞z2+izz3z3+1z3=limz→∞1z+i1z21+1z3=0+i(0)1+0=0…

Q: Show that limxy)H(-2,0) g(x, y) does not exist given that (x + 2)³y? (x + 2)6 + 3y4' 0, if (x, y) #…

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Q: lim x In(2x) x→0+

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Q: Show that lim e/n = 1 for any constant c> 1. Use the squeeze theorem (or any other method) to…

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Q: lim Vr In r.

A: To evaluate the given limit.

Q: lim zta sec z is A B 2 D nonexistent

A: Given query is to find value of the limit.

Q: 2n j3 lim - 24n n00 j=1

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Q: 2 i (z-i) lim Z. z²+1 1 O 2i 1 - 1

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Q: lim (VI+z- VF),

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Q: H-w in limits 3-3- 2.21 9 iaam ) Lim. 1- CosC2K) Sin (24) 2 - メャ3 2) Lim x2x -3 3) Lim (X+4}-16

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Q: 4) Let fn(x) = nx". (a) Show limn→∞ fn(x) = 0 for x ≥ [0, 1). 1 (b) Show limn→∞ ₁¹ fn(x)dx = 1.

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Q: Prove the limit doesn't exist

A: cc

Q: 12-x-x 1- Prove that lim -7 %3D x-3 V6+x-3

A: our objective is to find the limit

Q: lim -3- tan-() = lim-otan-ly %3D True False

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Q: x2-x-2 2x-4 2sin(6x–12) 3(x-2) Sx-1 Let f(x)* ,g(x) • and h(x) Vz+2+1 a. Find lim f(x), lim g(x),…

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Q: السؤال 1 lim iz?-z z-- z2-1 i o -i O 1 0 -1 O

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Q: lim, (V+x+1 - V+1) – Vx² x* + 1

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Q: EXAMPLES. Observe that iz +3 lim z→-1 z +1 z +1 =D0 since lim %3D z→-1 iz + 3

A: Now, limz→-1z+1iz+3=-1+1i(1)+3                 =0-1+3                 =0…

Q: lim Jx-S

A:

Q: A = lim %3D i = 1

A: Given that: The equation is f(x) = x 2 e x  ,     0 ≤x≤3 .

Q: 6n + 2 3. n→∞ 2n + 1 3. Using the definition (i.e. N, E), prove that lim

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Q: Find: lim→1+i z2 –z+1-i i z2-2z+2

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Q: 2 i (z-i) lim z →i z²+1 1 2i -1 | 1

A: Fourth option is correct.

Q: lim (2,y)+(0,0) In(2 + x² + y?) x2 + y?

A: We have to solve given limit:

Q: lim(z – 1) csc TZ Is

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Q: sin x 3. Prove that lim :1

A: To prove: limx→0 sin xx=1   Theorem Used: Squeeze theorem: If f(x) ≤ g(x) ≤ h(x) and limx→a…

Q: 3 a. lim うルーフ4 4- 1し 5- J3c+9 2-Jつい 5. しim うしー72 ル+2-Vt4んt4 C. Lim 2021+リ-26 ルーフ 20t ーて

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Q: 5. Prove that if lim g(r) = 0 then lim g(x) sin () = 0. %3D

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Q: lim does not exist NI

A: Given,  limz→0zz

Q: iz° + 1) lim z→-i\ z² 2 +1,

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Q: lim (2,1) (0,0) (1 + (x² + y²) In(2² + y²))

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Q: lim (z,y)→(0,0) (2+(x² +y²) In(x² + y²))

A: We have to find the limit of: lim(x,y)→(0,0) 2+(x2+y2)ln(x2+y2)

Q: z3+1 lim i z+1 z-

A: Note that, we solve the value of limit for following indeterminate form:    →00    →∞∞    →0×∞    →∞…

Q: lim %3D X2 -マメ

A: The solution for the above question is as shown below.

Q: lim (2 + (x + y²) In(x² + y³)) y)→(0,0)

A: Here, lim(x,y)→(0,0)2+x2+y2lnx2+y2

Q: (b) lim æ In(x² + y²) (x,y)→(0,0)

A:

Q: lim (x,y)(1,2) 2x +x*y-xy 4x2-y?

A: Given,          limx,y→1,22x3+x2y-xy24x2-y2

Q: I - 4 Prove that lim vI = 2 using the e-d-definition. Hint: Va - 2 = VT +2

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Q: If f(x) = xcosx then limf(x+h)-f(x) h COSX = ? %3D %3D h-0

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Q: lim (x,v)→(-2,0) g(x, y) does not exist given that Show that (x+2)³y² (x + 2)6 + 3y4' 0, if (x, y) #…

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Q: 2 -z+1-i Prove that lim =1- :l+i z- 2z+2 2

A: solution is given below

Q: X-2 lim = 2 a. By L'Hopital's Rule, x→² X²-1 1 1 lim = x→2 2x 4

A: