Calculate this limit using Maclaurin series: lim x rarr oo((x^3−x^2+2/x)e^1/x−sqrt(x^3+x^6))

vedentst9i

vedentst9i

Answered question

2022-11-18

Calculate this limit using Maclaurin series: lim x ( ( x 3 x 2 + 2 x ) e 1 x x 3 + x 6 )

Answer & Explanation

Laura Fletcher

Laura Fletcher

Beginner2022-11-19Added 22 answers

Step 1
I'm not entirely sure how the hypothesis about the expectation comes into play, but here's my solution. I'll simply use the Markov inequality for a non-negative r.v. and a > 0 in the form
E [ X ] a P ( X a )
with a = 2. Remembering that probabilities are non-negative and using Markov we have:
E [ X ] P ( X 2 ) + P ( X 2 ) P ( X 2 )
Edit after the comments about non-negativity:
I think I saw the light :D I use the extended form of Markov inequality for monotonically increasing function:
Step 2
Theorem
Let φ be a monotonically increasing function for x > 0 and let X be a r.v., a > 0 , φ ( a ) > 0. Then
P ( X a ) E [ φ ( X ) ] φ ( a )
In our case I'll consider the (somehwat artificial) function φ ( x ) := x + e x 1, which we can prove satisfy all the hypothesis with a = 2. Hence, since 1 / φ ( a ) 1 and using the bound for e X , we have
P ( X a ) E [ X + e X 1 ] φ ( a ) E [ X ] + E [ e X ] 1 E [ X ]

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