MAS110 Problems for Chapter 5: Limits and continuity
4
3
2
+x −1
for x 6= 1. Do a table of values of f (x) for some values of x very close to
1. Let f (x) = x −xx−1
1. Prove algebraically that limx→1 f (x) is what it appears to be.
3x−7
. Do a table of values of f (x) for larger and larger values of x. Prove algebraically
2. Let f (x) = 5x+3
that limx→∞ f (x) is what it appears to be.
3. Evaluate x1/x for larger and larger values of x to see if limx→∞ x1/x might exist. Do the same for
limx→∞ (1 + 1/x)x .
4. Graph the function


3 − x, when x < 1,
f (x) := 1,
when x = 1,


x − 1, when x > 1.
Find limx→1− f (x) and fx→1+ f (x). Does limx→1 f (x) exist? Is f continuous everywhere?
5. The function f : [−2, 2] → R is defined as follows:

1 + x, if



1 − x2 , if
f (x) =

1,
if



2x,
if
−2 ≤ x ≤ −1,
−1 < x < 1,
x = 1,
1 < x ≤ 2.
Sketch the graph of f . Which of the following statements are true?
(a)
(b)
(c)
(d)
(e)
limx→1+ f (x) = f (1).
limx→1− f (x) = f (1).
limx→1 f (x) = f (1).
limx→−1− f (x) = f (−1).
limx→−1 f (x) = f (−1).
Discuss the continuity of f .
6. The function f : R → R given by
(
cx2 + 2x, if x ≤ 2
f (x) =
x3 − cx, if x ≥ 2
is known to be continuous on the whole of R. Determine c.
7. f and g are continuous functions. If g(0) = 5 and limx→0 (3f (x)g(x) + 2f (x)) = 34, what is
f (0)?
1
8. Evaluate the following limits.
x4 − 1
x2 + 2x + 1
(ii)
lim
x→1 x3 − 1
x→−1
x4 − 1
1
− 13
(a + h)3 − a3
x
(iii) lim
(iv) lim
x→3 x − 3
h→0
h
√
√
√
1+h− 1−h
x2 + 9 − 5
(v) lim
(vi) lim
x→−4
h→0
h
x+4
√ √
√
√
(For (v) and (vi) try to exploit ( a + b)( a − b) = a − b.)
(i)
lim
9. For x ∈ R, let bxc is the largest integer not exceeding x i.e. bxc ∈ Z is the unique integer such
that bxc ≤ x < bxc + 1. Using the sandwich rule, evaluate the limits
bxc
x→∞ x
lim
10. Show that lim x2 sin
x→0
and
bxc
.
x→−∞ x
lim
1
= 0 by using the sandwich rule.
x
sin(7x)
sin x
by using that lim
= 1.
x→0 x
x→0
x
11. Find lim
12. Show that
1 − cos x
sin2 x
=
x
x(1 + cos x)
1 − cos x
= 0.
x→0
x
and deduce that lim
13. Use the Intermediate Value Theorem to show that the equation sin x = x2 − x has a solution in
(1, 2).
14. Use the Intermediate Value Theorem to show that the equation
(2, 3). Find the root correct to 2 decimal places.
√
2
= x − x has a solution in
x
15. Is there a real number that is exactly 1 more than its cube?
!
√
√
√
√
√
√
x + x + x − x . [Hint: use a − b = (a − b)/( a + b) and then
16. Evaluate lim
x→∞
√
divide numerator and denominator by x.]
r
q
2