Let fn(x)= exp (-x^)+1 for all xE Lo,1] for allneN* Cer Pn (x)= exp (-x^)+) for all x e [o,l] Por allneN* then which is true ! 9 (Fn)n Converse s pointwiely to fiCo,!]R where flx)= 2 2) (fn )n is inerease and lim fn lx)dHL=2 3) (Fn)n converses unifurmly [o,l] 4) (Fn)n is decreasing and lim fn(w) duL lim fnx)deL

Q: One of the following ODE is a linear DE of second order O x³y"-3y'=e*+xy O None of these O…

A: We are given some ODE and out of which we have to identify which ODE is linear DE of second order.…

Q: 4) Define f: N→R as follows: f(1) = 3, f(2) = 3 and for n > 3 f(n) = f(n- 1) + f(n- 2) %3D %3D 2'…

A: (4) Given data are                                 f:N→Rf(1)=3, f(2)=32 and for…

Q: (8) Let X-b(n,p), with E(X) = 6 and var (X) = 4.2, then P(X ≤ 6) = 0.608.

A:

Q: 4 G DI -1 Ic < 2/2 - NI 26 letters 10 digits 0-9 digits = 10 digits { possible numbers} 10 x 10 x 10…

A:

Q: Find f'(x) if f(x) = - 32 - 342 6x² 322 x 8x³+x4 82 8a+a 3 æ3 _ 113 2

A: Given query is to find indicated derivative.

Q: k n = ( 2 ) · ( 5 ) = ( 7 ) · (2 – 7 ) for k k-r Prove Newton's Identity: for n ≥k≥r≥ 0.....

A:

Q: (a) Suppose X U (0,n), where n is a positive integer X-[X] U (0, 1). (b) Use the result of (a) to…

A: De Moivre's theorem gives an axiom for computing the powers of complex numbers. We first gain some…

Q: Show that f (x) = is not uni form continuous on (0,3) -3

A: To prove that 1x-3 is not uniformly continuous on 0,3

Q: 1. Fin d fx) for each of th FunCtions. 54ow your work,Gut you notneed to simplif y answer Sint PartA…

A: A)f(x)=sin xxusing quotient rule;f'uv=v u' -u v'v2u=sinx, v=xf'(x)=xddx(sinx)-sin(x)ddx(x)x2f'(x)=x…

Q: Prove that Hence deduce that L{x"; p} r(n + 1) pn+1 " if n>-1, p > 0. L{x-¹/2; p} = √√(π/p),p> 0

A:

Q: Let α = 2 + √2 and β = 2 −√2. Then n ∈ N implies (a) α^n + β^n ∈N and α^n + β^n = [α^n] + 1, (b)…

A: Given:                    α=2+2 and β=2-2 To prove: ∀ n∈ℕ αn+βn∈ℕ and αn+βn=αn+1 limn→∞αn+αn=1

Q: (응) cost/ of the funcfion 4) EStimate the vawe of In osing a linearization f(e)> In (x) cos² (x).

A: Given: The function fx=lnx cos2πx To estimate: The value of ln65cos26π5 using linear factorization.

Q: Q6. (i) Given Iog(itt) = tE+ t'-t"+ -.., valid for IEl<1, determine the power serhes for log (+2x )…

A:

Q: dx. J x² + 2x + 5 x+1 11 arctan x+1 11 ;+ 2x – In [/r² + 2x + S- *+ 2x + In Vx? + 2x +5+, arctan 2 2…

A:

Q: If 40 f(x) dx = 13, then compute the value of the following definite integral. 20 8 fºr(² f(5x) + 4…

A: By the answering guidelines of Bartleby, we can answer only first question from multiple questions,…

Q: Determinewhetherthefollowingsequencestendtoinfinityandgiveafullargument establishing your…

A: To determine whether the sequences tend to infinity, we need to analyze their behavior as n…

Q: Let Ci and co be arbitrany canstants. Geneial solutian to the hamogencus dfferenti al oquation 1692…

A: Given differential is Cauchy Euler's differential equation

Q: Show that f is continuous on (-0,00) sin x if x 4 V

A:

Q: ( b) For n=1,2,K , show that (x² –1)y"*) +2(nx + p)y") +(n² –n)y"-1) =0 where %3D d" y p") = (Hint:…

A: * Given that y = x+1x-1pa dydx = px+1x-1p-1ddxx+1x-1    dydx = px+1x-1p-1x-11-x+1.1x-12   dydx =…

Q: Find the arc length (s) of the curve 7(t) = (e*, 4e", - 2et) for 0 <t < In(5) |

A: we are given the curve  r→(t)=e5t,-4e5t,-2e5t =e5ti-4e5tj-2e5tkwhere 0≤t≤ln5

Q: Express the given integral f, 2x dx as the limit of Riemann sum. a. 2x dx = lim E-(1+) G+1)* E- d.…

A:

Q: Q7/ * find f`(2/3), where f(x) =x tan-'(x/2) T/3+v3/4 O H/3+v3/4 O T/3+v(3/4)

A:

Q: If you are asked to find L4 for f(x) = edz) + In(x) on the interval [2,10], what would the value of…

A: From the given problem:   fx=e3x2+lnxa,b⇒2,10   As we know that the expression for the step size is:…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.