Real Analysis Answers

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?

Prove that a strictly decreasing function is always one-one

Find the following limit

Lim x tengs to 0 1-cos x^2/x^2 sin x^2

1. Use mathematical induction to show that n! ≥ 2(n-1) for all n ≥ 1.
2. Use (1) and the definition of a Cauchy sequence to show that S_n = ( 1+ 1/2! + 1/3! + ⋯ 1/n!) is Cauchy sequence.