Arrange the following in order of increasing size in the asymptotic limit that δ → ∞:

* State the definition of what it means to write f(x) ∼ g(x) as x → a. Then, construct an asymptotic behavior for the following functions in the given limits:
* 4 cosh x as x → ∞.
* x 2sin x as x → 0.
* cot x as x → π.
Consider the polynomial x 7+Ex3 = 2 which has one purely real solution. Find an approximation of this root in the asymptotic limit that E → 0. Find as many terms as you need to ensure that the solution is accurate to 4 d.p when E = 0.01. Comment on how you are assessing whether the requisite accuracy has been obtained.
The Lambert-W function, W(x), is defined by solutions to the equation x = W(x) exp. Given that W(1) = 0.567143, use perturbation methods to find a two-term approximation of W(1.01).
Consider the initial value problem x 0 + Ex = t with x(0) = 2. Find a two-term approximation of the solution in the asymptotic limit that E → 0.
The error function is defined via the integral

Use repeated integration by parts to find a three term approximation of the value of the error function in the limit that x → ∞. Use this to approximate the values of Erf(1), Erf(2) and Erf(3). How does your values compare to the exact values of Erf(1) = 0.842701, Erf(2) = 0.995322 and Erf(3) = 0.99978 to 6 d.p.?

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