Real Analysis Answers

1. a). Write down the definition (ε-δ language) of limx→x0 f(x) = L. b). Show that limx→+∞ cos x does not exist.

Let f be a differentiable function on [a,b ] and x belongs to[a,b]. Show that, if f'(x)=0 and f''(x)>0, then f must have a local maximum at x.

Let f :[0,1] tends to R be a function defined by f(x)=x^m (1-x)^n, where m,n belongs to N.Find the values of m and n such that the Rolle’s Theorem holds for the function f .

Find the following limit x tends to 0 (1-cosx^2/x^2 - x^2 sin x^2)

Determine the local minimum and local maximum value of the function f defined by f(x)=3-5x^3+5x^4-x^5

The function f, defined by f(x,y)=x^3+xy+y, is inegtrable on [1,2]×[1,3]. True or false with full explanation.

Give an example of an series Σan such that Σan is not convergent but the sequence (an) converges to 0

Check, whether the collection of G, given by

G'= { ] 1/n+1,1/n [ : n∈N } is a open curve of ]0,1[

Test the series

∞

Σ(-1)^(n-1)× sinnx/n√n for absolute and

n=1 conditional convergence

If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuous compounded account?