Question: Let h(x) := e-1/x2 for x 0 and h(0) := 0. Show that h(n)(0) = 0 for all n N. Conclude that the remainder term

Let h(x) := e-1/x2 for x ‰ 0 and h(0) := 0. Show that h(n)(0) = 0 for all n ˆˆ N. Conclude that the remainder term in Taylor's Theorem for x0 = 0 does not converge to zero as n †’ ˆž for x ‰ 0. [By L'Hospital's Rule, 0 for any k ˆˆ N. Use Exercise 3 to calculate h(n)(x) for x ‰ 0.]

lim h(x)/x*

Step by Step Solution

★★★★★

3.35 Rating (161 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Use Induction to show hn0 0 for n N If x 0 then hnx is the sum of terms of the form e 1x2 x k therefore if h n 0 0 then hn ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

829-C-D (643).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Assume that the consumption set X is nonempty, compact, and convex. Let denote the budget correspondence. Choose any (p, m) P such that m > minxX mi=1 Pixi, and let T be an open set such that X(p, m)...

Let the degrees of the polynomials P(z) = a0 + a1z + a2z2 +· · ·+ anzn (an 0) And Q(z) = b0 + b1z + b2z2 +· · ·+ bmzm (bm 0) Be such that m ¥ n + 2. Use...

Let Show that if d = a11a22 - a21a12 0, then A=[2:22] a2i 22 a22 -a12 d -a a 421 41)

Least Squares Approximation Hector D. Ceniceros 1 Least Squares Approximation Let f be a continuous function on [a, b]. We would like to find the best approximation to f by a polynomial of degree at...

Kindly please digitalized the answer 1- Integrate the following. 1. fx'dx 2. JT 3. S(x2 - 2x + 5) dx 4. S(x + 2)5 dx 5. S1+ y2 2ydy 6. S cos(70 + 5) do 7. S ( x2 + 2x - 3)(x + 1) dx 11- Find the Area...

KINDLY PLEASE DIGITALIZED IT 11- Find the Area between the curve y=x(x-3) and the ordinates x=0 and x=5. Note: (hint) The Graph shows 2 regions. Get first the area of each region before adding it...

2. Inferences about the difference between two population means - Independent samples, population standard deviations known Consider a pool of home mortgages. Prepayments of mortgages in the pool...

just to let you know the first 3 parts are already answered part A,B,C part A is the picture under the first picture part B are the two pictures directly after Part A picture and part C is the Last...

Could you show explanation and working for this example Let h > 0 and let P, (0, 0), Q, and be the three points on the pianar curve y = x2/2 at x = h, x = 0. and x = h, respectively, and let the...

Solve problem 1.16 from the book Ordinary differential equations and Dynamical systems by the author Gerald Teschl . You need to use implicit theorem or inverse function theorem of Differential...

Using EES (or other) software, study the effect of varying the compression ratio from 5 to 10. Plot the net work output and thermal efficiency as a function of the compression ratio. Plot the T-s and...

ABC Company had purchased a $7,500, 24-month insurance policy for a vehicle on November 1 with the coverage effective date to be the same date as the purchase. Assuming only the only associated entry...

Commercial banks are required to include swap risk exposure when calculating risk - based capital requirements. True false question. True False

Which type of marketing involves creating and distributing content that is valuable and relevant to a companys target customers? a. Website marketing b. Blog marketing c. Email marketing d. Content...

Write the expression that represents the area of each composite figure. Then simplify to find the total area. 12 kilometers ( 5) kilometers Rectangle (5 1) kilometers 4 kilometers Rectangle i

Simplify the following 761268x + 5922 - 2960

Find the perimeter of each figure. See Example 14. 1. 2. 3. x feet 7 feet 3x feet 5x feet 4x feet 12 feet 3z meters 1 meter 1 meter 5z meters

Cash fow to the waman under an interest-only loan, in which Poryi will pay the annual intecest expense each year and pay the principal back at the end of the contract What is the amount of payment...

A project will produce operating cash flows of $45,000 a year for four years. During the life of the project, inventory will be lowered by $30,000 and accounts receivable will increase by $15,000....

30. Consider the market for one-year discount bonds with standard face value of $1,000. Assume that supply and demand curves in this market are both linear, and Q$ = P 700, while demand curve passes...