Exercise 1. Find the Taylor expansion of the function S(z) = Vsin r around the point r = 0 up to the term z3. show that there exists the limit lim, 40 f'(x) write the Taylor expansion of sin t near t = 0, substitute t = r* • write the Taylor expansion of f(x) near r = 0

Q: 6. Name three distinct geometric situations in which the graph of a function would fail to be…

A: We have to describe the three distinct situations geometrically, in which the graph of the function…

Q: 3. Use a limit to find f'(4) for f(x) = x²-x.

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Q: how did you get y^2-4xy to y-4x when there are three y's on top I input the numbers and got 0 which…

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Q: Show limx→∞ xne−x = 0 for all n = 0, 1, 2, . . . . To get started notice that when x ≥ 0, all the…

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Q: (fif2..f)is di fferentiable at x, proofthat ... by using f(x)– f(*.) f (x)=lim, x-x,

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Q: Let U = {1,2,3,4,5), A={1,2,5}, B = {1,4,5), and C= {1,2,3}. (a) Find A U (BUC). (b) Find (A U B) U…

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Q: limx→∞ 2*+1 + 3* - 2 -1 - 3*+2

A: To determine the value of limx→∞2x+1+3x-22x-1-3x+2, divide the numerator and the denominator of the…

Q: ,3x lim x→0 CO cosh(Зa)

A: By substituting cosh(3x) in givenimit we can get the required value of Function.

Q: Assume that f : R → R is defined via the assignment rule f( x ) = e^(2x) . We will use the notation…

A: Step:-1 Process for induction Method:- To show any statement true using induction, we start with…

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: How do we calculate the limit of limx→0  (sin(x))/(9x3), by using the Taylor approximation

A: limx→0  (sin(x))/(9x3)

Q: 1 S(t – 4t) dt

A: Given that: The limit is limN→∞ ∑i =1N  iN2 - 4iN×iN .

Q: what is the limit of (cos(x))^4/x^2 as x approaches 0 from the positive side?

A: Given,

Q: Compute f'(x) using the limit definition. f (x) = 3x − 7

A: To find the derivative: f(x)=3x-7 Solution: By definition; f'(x)=limh→0f(x+h)-f(x)h Given,f(x)=3x-7…

Q: Why does the limit as x goes to 0 of cos(1/x) not exist? Can use two different sequences:…

A: Solution

Q: Compute f'(x) using the limit definition. f (x) = x−1/2

A: Given function is fx=x-12

Q: 3. Use the limit definition of the derivative to find f'(x) for f(x) = 7x² + x.

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Q: f, there is a sequence (zn) in C - {1¹} such that : lim z, = 1, but f(zn) n n→∞ has no limit in C U…

A: Take sequence z(n) as Z(n) = 1 + {[(-1)^n)]/n}So that as n goes to infinity  Z(n) goes to 1 .

Q: Show that an is decreasing and bounded from below where an+1 = (1+a^2n)/3 and a1 = 1. Then determine…

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Q: What is the limit as t goes to 0 of    4 tan^-1(t)

A: Topic:- limit

Q: Find the limit as a→5 for ((1/a)−(1/5)/ (a−5))

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Q: The limit lim h→0 f(x) = cos(h) - 1 h represents f' (a) for some function f and some number a. Find…

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Q: Suppose that f satisfies the equation f(a +b) = f(a) + f(b) + a'b+ ab (E 1, b €R. Suppose further…

A: given :f(a+b)=f(a)+f(b)+a2b+ab2→(1)limx→0f(x)x=1 →(2)a.) put a=b=0 in…