ASSIGNMENT-3 (LIMIT AND CONTINUITY)

Q1:- Find ε and δ such that limn→1x

x+1= 1

2.

Q2:- Find ε and δ such that limx→c

√x =√c.

Q3:- Find ε and δ such that limx→c x3 = c3.

Q4:- Find ε and δ such that limx→1x

x+1= 1

2.

Q5:- Find ε and δ such that limx→c1x= 1

c.

Q6:- Show limx→0 sgn (x) does not exists. Where sgn is a signumfunction.Q7:- Show that limn→0 sin

(1x

)does not exists in R.

Q8:- Find the following limits (Hint:- Use squeeze theorem)

(1) limn→∞1+|sin(n)|

2n.

(2) limn→∞(2n−53n+1

)n.

Q9:- Let the sequence 〈xn〉 converges to x. The show that the sequence〈|xn|〉 converges to |x| . That is, if limxn = x then lim |xn| = |x| .Q10:- Let A be a non-empty subset of R. De�ne the function f :

R→ R byf (x) = inf {|x− a| : a ∈ A} .

Then show that f is uniformly continuous.Q11:- Let f be a dirichelt's function de�ned as

f (x) =

{1 if x is rational0 if x is irrational

Then show that f is not continuous.

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