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Show that the limit  lim x approaches infinity [(1+x/2-sqrt(1+x)) / x^2] exists and find its value. I have that the Taylor polynomial T2[sqrt(1+x),0] = 1 + (1/2)x -(1/8)x^2. And an expression for the remainder R2(x), sqrt(1+x)=T2(x)+R2(x) = 1/(16sqrt(1+c)^5) x^3.

Show that the limit  lim x approaches infinity [(1+x/2-sqrt(1+x)) / x^2] exists and find its value. I have that the Taylor polynomial T2[sqrt(1+x),0] = 1 + (1/2)x -(1/8)x^2. And an expression for the remainder R2(x), sqrt(1+x)=T2(x)+R2(x) = 1/(16sqrt(1+c)^5) x^3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Show that the limit 

lim x approaches infinity [(1+x/2-sqrt(1+x)) / x^2] exists and find its value.

I have that the Taylor polynomial T2[sqrt(1+x),0] = 1 + (1/2)x -(1/8)x^2.

And an expression for the remainder R2(x), sqrt(1+x)=T2(x)+R2(x)

= 1/(16sqrt(1+c)^5) x^3.

Expert Solution
Step 1

Given limx→∞1+x2−sqrt1+xx2.

We have to show that the given limit is exists and find the its value.

Given that the Taylor polynomial T2sqrt1+x,0=1+x2−x28

If we substitute the limit then we have of the ∞∞, which is indeterminate form.

 

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