4) Prove that if lim(x,) = x and if x > 0, then there exists a natural number K such that x, > 0 for all n > K.

Q: Using the definition of limit (so, without using Arithmetic of Limits), show that    a) limn→∞ (4 +…

A: Given: a) limn→∞ 4+n2n=12 b) limn→∞ 2n+3n+1=0 Prove that the given limit without using the…

Q: b) Solve the limits of the following expressions. i) lim *-1 x→3 x+3' ii) x²+6x+9 lim x→-3 x+3

A:

Q: 2. Question: Prove that the limit does not exist x² + 7x + 9 lim X→-5 x2 - 25

A: Given: limx→-5x2+7x+9x2-25 for evaluating given limit, we put x=-5 in given limit

Q: lim x² ln x x-0*

A:

Q: 5. Prove that lim x+0+ x3 (HINT: This proof needs a B–8 type argument.) = 0 and lim x→0- x³ ||

A: please see the next step for solution

Q: Let a be any positive real number. Then * 1 lim In a – 1 True False

A:

Q: 2) Evaluate the limit, if it exists. x²-x-12 a) lim X--3 x+3 b) lim x-0 √1+3x-1 c) lim (₁-²₁) X-1 11…

A: For (a),      limx→ -3x2-x-12x+3=limx→ -3x-4Plug in the value x=-3=-3-4=-7

Q: Pind the limit lim 848 (√7x² +10 -√7x²-3)

A: Recall the formula: limx→∞ c=c and limx→∞ ax2±b=∞

Q: 3) Evaluate the infinite limits. a) x+1 lim x→1+ x sin éx 1 b) lim x-3-(x-3)5

A:

Q: Prove the limit statements

A: To prove: =lim     x→04-x=4-0=4=2=RHSHence proved To prove this, we will start from LHS and will try…

Q: Suppose k element R is a constant. Prove that. lim (x - - > a) k=k

A: Definition of a Limit: The limit of f(x) as x→a is denoted by limx→af(x)=L if, for every positive…

Q: x 3 - x² 4) lim x→1 X-1

A:

Q: 47) lim √√3x-2 x-1 48) lim X->>0 x3 + 12x² - 5x 5x

A: To find the limit of a function f(x) when x approaches k, substitute x= k in f(x)

Q: Compute lim 1+-

A:

Q: 3* – 1 compute lim V1+x -1

A:

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

A: our objective is to find the limit

Q: Prove that lim n = p.

A:

Q: 2. Use the Squeeze theorem to show that lim x-0 ² co(+1)²

A:

Q: 1 Assume that lim xn +oo. Prove that lim n-00 n +o0. n-00 k=1

A: Given limn→∞xn=+∞

Q: 11. lim x→ー4 x2-16 2x2+9x+4 z2-36 12. lim Z--16 Z-6

A: (a) We have the given limit problem as limx→-4x2-162x2+9x+4 On calculating the factors of numerator…

Q: Find the limits t +3 a) lim (4x3 – 3x² + x – 85); b) lim, t+2 t+1 ; c) lim x +0 1-V1+7x ; d) lim (7…

A: We can solve these problems in this way

Q: (2n+1)!+(2n+2)! )! (2n+3 lim

A:

Q: Q.2 Show that r(ax) _ 1 lim x0 (x)

A: to show  limx→0ΓaxΓx=1a   ...i as we know Γn=Γn+1n using this formula we get Γx=Γx+1xΓax=Γax+1ax

Q: 11) Evaluate the limits. Be sure to show all work. a) lim r0 2x+6x -1 5x -8x+1 2 sin x b) lim-

A:

Q: 1 -1 B lim² =? Limit mevcut değildir D E

A: We will solve the following

Q: 5) Evaluate each limit, if it exists. Vx+1-1 lim х2-2х-63 lim x→-7 a) b) x²-49

A: The given problem is to evaluate the given following limits, We can use L-Hopital rule to evaluate.…