202122: l'Hôpital's rule, integrals. You are given that limx→o (x log x) = 0 and that limx→0 logx = 0. Let m, n € 4. a) Use L'Hôpital's rule to show that lim [xm (log x)"] X→0 m n x→0 lim [xm (log x)"+1]. Hence, by induction or otherwise, show that lim [x (log x)"] = 0. X→0 b) Manipulate xm (logx)" dx into an integral of similar form but with a reduced power or powers in the integrand. c) Hence compute ¹ x6(log x)5 dx.

Q: Prove that f(x)=log(x)-x^2 +1 has exactly two roots namely x=1 and x=0.45

A: Function: f(x)=log(x)-x2+1 If it has two roots x=1 and x=0.45 then f(1)=0 and f(0.45)=0.

Q: (log x, 0 < x < 1 a) g(x) = x21 i. lim g(x) x-1 ii. lim g(x) iii. lim g(x) X-2+

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Q: Deduce that log (1 + x) 0. If fix) = log (1 + x), x > 0, uning Maclaurin's theorem, show that for…

A: Given that: fx=log1+x, x&gt;0 To show  log1+x=x−x22+x331+θx3 for 0&lt;θ&lt;1 and log1+x&lt;x−x22+x33…

Q: 3. a) c) 4. a) Which statement is true of the following logarit Domain: {y ER} Domain: {x> 1}…

A: For a logarithmic function  to be defined  for .The range of a defined logarithmic function is .

Q: limx→0 limx→1 log(1+x) X log x 1 – x

A: We know the expansion of log(1+x) is: log(1+x)= x -x2/2 +x3/3 -x4/4+......

Q: limx→0 (x log x) = 0 and  limx→0 1/log x = 0. Let m, n ∈ N. i. Using L’Hˆopital’s rule, prove…

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Q: -a' 1-cospa O 1) Find the limt of logx-¬a . And Also Find the limt of loge¬o x"-a" 1-cosq6

A: When applying limit the value is 0/0 then aaply d'hopital rule I.e differentiate numerator and…

Q: 7. Solve the following. Check for extraneous roots. a) log₂ (x + 25) − log₂ (x - 1) = 3 b) log x +…

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Q: Note that lim (1 - 7x)1/x is indeterminate of type 1°. Using logarithms, we have In(1 – 7x). As x-…

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Q: In x

A: A detailed solution is given below

Q: 8. Solve for x in: :log (x² + x - 1) = log (x)

A: given,  logx2+x-1=logx

Q: Evaluate lim (sin z)an. 10.

A: **As per our guidelines we are allowed to answer only one question at a time. Kindly please re-post…

Q: 3"x2-4x+7 3x2-5x-1 Find the value of a so that lim loga Hint lim flg(x)] = f lim g(x) %3D O 4/5 01

A: Given

Q: 95. Use logarithms to prove that for any real number r, (1+)* =e". lim

A: Consider, limx→∞1+rxx=elnlimx→∞1+rxx     ∵elnx=x=elimx→∞ln1+rxx=elimx→∞x·ln1+rx…

Q: 2. Solve: -) 35x-1 = 74x+1 ) Log(x)=√4Log(x) + 5

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Q: 2COS X-I Sin x, Yna1 = 2cosxy, – Yn-1+ n21 and 0<xs. (i) Use mathematical induction to prove that…

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Q: lim ln(7-x) x →7

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Q: 5) Consider f(x) = log (x + 3) a) Solve f(x) = 0 by working backward.

A: f(x)=log⁡(x+3)

Q: Let f(x)=hx  and   g(x)= hsin^2x a. Evaluate f(tan2pi) b. Solve h1/2=hcos^2x  exactly for x in the…

A: Given functions are a) We know that tan2pi=0 Therefore,

Q: 5. Approximate 8 log(2023) and Led xex using the Romberg algorithm. Discuss which integrals you…

A: Richardson extrapolation is a numerical technique used to improve the accuracy of an approximation…

Q: lim,-0 e" = ?

A: Consider the given limit limx→-∞ex.

Q: 10. Prove that (log2 3)-1+ (log4 3)-1 < 2.

A: Simplify the powers of the term (log23)-1+(log43)-1 by writing the terms as fractions.…

Q: (19 In Matlab the folCan be written as: g log 10(exp(1)^(1/3)-sinh(4*x^5)/2)lowing equation' 19) In…

A: First I refresh you about the mathematical symbols in  Symbol Operator ^ Raise to the power *…

Q: -e lim te Int - 1

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Q: 5. Determine if the following are true or false. loga logb _a) log = logx – 3 logy d) log %3D 3 _b)…

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Q: (log n)* lim

A: Explanation and solutions is given below....

Q: 8) Solve for x when log10¹ = x a) log 8 100,000,000 b) log 10 c) 8 d)

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Q: ompute, if it exists, lim 1-0 log(1 + r) + (1²/2) - sin(x) (³²-1)

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Q: 6) prove that cost '2 = log (2 + √2²-1).

A: To prove that cosh-1(z) = log(z + sqrt(z2 - 1)), we can start by defining the hyperbolic cosine…

Q: Q9

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Q: 15. Solve each equation for x. b) 9*-1 = 31+ c) In(x+4)+ln(x-2) = 0 l+x a) e2" = e¤²–3 d) log +log 3…

A: (a)

Q: properties of logarithms to

A: Use properties of logarithm to expand the given expression Explanation:(1) We know that by quotient…

Q: * limt (x-sinx)\ x+1 10 log x^2+3 O 0.5

A: