Advanced Calculus, Assignment 1

1. Let f : R→ R and c ∈ R. Show that

limx→c

f(x) = L if and only if limx→0

f(x + c) = L.

2. Let {yn}n∈N be a sequence in a metric space Y . Define a function on theset

D =

{1

n: n ∈ N

}by f

(1n

)= yn. Show that f has a limit at 0 if and only if {yn}n∈N is

convergent.

3. Using the definition of limits (the ”epsilon-delta” definition), verify thefollowing limits:

(a)limx→3

(x2 − x) = 6

(b)limx→c

x = c for all c ∈ R

(c)limx→4

√x = 2

(d)

limx→1

x

1 + x=

1

2

4. Use the sequential criterion of limits to establish the following limits:

(a)

limx→2

1

1− x= −1

(b)

limx→9

1√x

=1

3

(c)

limx→0

x2

|x|= 0

(d)

limx→1

x

1 + x=

1

2

1

5. For p ∈ R, let f(p) = (2p+ 1, p2) ∈ R2. Then clearly f : R→ R2. Use thedefinition of limit to prove that

limp→1

f(p) = (3, 1)

with respect to the Euclidean metrics on each space.

6. Let c ∈ R and let f : R→ R be such that limx→c(f(x))2 = L.

(a) Show that if L = 0, then limx→c f(x) = 0.

(b) Show by example that if L 6= 0, then f may not have a limit at c.

7. Determine the following limits and verify your result:

(a)limx→1

(x + 1)(2x + 3)

(b)

limx→1

(x + 1)

(2x + 3)

(c)limx→1

(x + 1) + (2x + 3)

(d)limx→1

(x + 1)− (2x + 3)

8. Consider the functions f, g and h defined by

f(x) = x + 1 g(x) = x− 1 h(x) =

{2 if x 6= 1

0 if x = 1

(a) Find limx→1 g(f(x)) and compare with the value of g(limx→1 f(x)).

(b) Find limx→1 h(f(x)) and compare with the value of h(limx→1 f(x)).

9. Let f : R → R and g : R → R be such that limx→c f(x) = L andlimx→L g(x) = M . Prove that limx→c g(f(x)) = M .

10. (a) Give an example of a function that has a left-hand limit but not aright-hand limit at a point.

(b) Give an example of a function that has both left- and right-handlimits, but not a limit at a point.

11. Define f : R→ R by

f(x) =

{x− 1 if x < 0

x + 1 if x ≥ 0

2

(a) If {xn}n∈N is an increasing (decreasing) sequence which convergesshow that {f(xn)}n∈N is also an increasing (decreasing) sequencewhich converges.

(b) Find an example of a sequence {yn}n∈N which converges but {f(yn)}n∈Ndoes not converge.

12. Find an example of a function f : R→ R, a sequence {xn}n∈N and a pointc so that f(xn)→ f(c) but xn 6→ c.

13. Prove that the following limit does not exist

limx→0

f(x)

where f(x) = 1 if x is a rational number and f(x) = 0 if x is irrational.

14. Prove that the following limit does not exist:

limx→0

1

x

15. Let f(x) =

{1 if x ∈ Q0 if x ∈ R \Q.

(a) Does limx→0 f(x) exist? Explain why.

(b) Does limx→0 f |Q(x) exist? Explain why.

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