Real Analysis Answers

Are the following statements true or false? Give reasons for tour answers.

a) −2 isalimitpointoftheinterval ]−3,2].

b) The series (1/2) - (1/6) + (1/10) (−1/4) +.... is divergent.

c) The function, f (x) = sin²x is uniformly continuous in the interval [0,π].

d) Every continuous function is differentiable.

e) The function f defined on R by

f(x)= {0, if x is rational and 2, if x is irrational

Is integral element in the interval [2,3].

Show that the series n=1∑∞ x/(1+n²x²) is uniformly convergent in [α,1] for any α>1.

Show that the sequence (a_n ),where a_n [ n/(n²+4) ] where is monotonic. Is (a_n ) a Cauchy sequence? Justify your answer.

Check whether the function f given by:

f (x) = (x − 4)³ (x +1)²

has local maxima and local minima.

Show that on the curve y = 3x² − 7x + 6 the chord joining the points whose abscissa

are x = 1 and x = 2, is parallel to the tangent at the whose abscissa is x= 3/2

Using the sequential definition of the continuity, prove that the function f , defined by:

f (x) = - 3, if is rational

f (x) = 3, if is irrational

is discontinuous at each real number.

Let {a_n} be a sequence defined as a₁ =3, a_n+1 = (1/5)a_n ,show that {a_n } an converges to zero.

For the function, f(x) = x² −2 defined over [1,5], verify : L(P, f ) ≤ U(−P, f )

where P is the partition which divides [1,5] into four equal intervals.

Show that

i) lim x→∞ [ (x-3) / (x-1) ] ^x = 1/e²

ii) lim x→5/3 1/ (3x+5)² = ∞

Give an example of an infinite set with finite number of limit points, with proper

justification.