Show that 

Check whether f is uniformly continuous on [− ]1,1 or not.

 Find the radius of convergence of the series .

For x∈ ]2,0[ and n∈N, define f_n (x) = 3 x² + 2x/n. Find the limit function ' f ' of the sequence (f_n ) _n∈N , Is f continuous? Check if  and  are equal or not.

Apply the Cauchy’s integral test to evaluate:

Let f : R → R be a function defined by:

Check whether f ′ is continuous on R.

 Prove that between any two real roots of ,2 e cos2x = x there is at least one real root of e^x sin

Let x and y be two real numbers such that x < y.Show that there exists an irrational number λ such that

x < λ < y.

Write the following statement, and its negation, using logical quantifiers. Also interpret its negation in words.

∃ x∈R such that x-1/3 > 0.

 Show that [a,∞)is closed set.

Find the supremum and infimum of the set:

Disprove the statement:

by providing a suitable counter-example.

 Prove that a function f : S → S (where S is a finite non-empty set) is injective if it is surjective.

Test the following series for convergence:

 Show that the set:

is not closed.

 Prove that the sequence (a_n)_n∈N, where a_n = 3² / x² + 2^{2 ,} is Cauchy.

Test the following series for convergence.

(i)  

(ii) 

 Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:

i) Every infinite set is an open set.

ii) The negation of p^~qis p→q.

iii) -1 is a limit point of the interval ]-2,1].

iv) The necessary condition for a function f to be integrable is that it is continuous.

The function f: R→R defined by f(x) = |x-2|+|3-x is differentiable at x = 5.

यदि ?1 , ?2 ∶ ? → ? एक परिमित विमा सदिश समष्टि ? पर स्वसंलग्न संकारक है, तो T₁ + T₂ भी स्वसंलग्न है।

यदि एक ? × ?, ? ≥ 2,, वर्गीय आव्यूह विकर्णनीय है, तो इस के अल्पिष्ट बहुपद और अभिलक्षणिक बहुपद बराबर है।