DEPARTMENT OF MATHEMATICAL SCIENCES, IIT (BHU)
ODD SEMESTER 2023-24
MA101 – ENGINEERING MATHEMATICS 1
TUTORIAL SHEET 4
Exercise 1: Using the precise definition of limit, prove that
3
(i) limx→2 xx2 −4 = 4.
+1−1 5 −1
(ii) limx→0+ e x + 1 = 1.
(iii) limx→β f (x) = β α where α, β > 0 and
(
xα if x 6= β,
f (x) =
2 if x = β.
Exercise 2: Consider the function f : R+ → R defined by
(√
x − a if x ∈ R \ Q+
f (x) = √
b − x if x ∈ Q+ .
Show that
(1) limx→ 2 f (x) exists.
( a+b
2 )
2
(2) limx→c f (x) does not exists for any 0 < c 6= a+b
2
.
Exercise 3: Suppose a function f : (−a, a) \ {0} → (0, ∞) satisfies
1
lim f (x) + = 2.
x→0 f (x)
Show that limx→0 f (x) = 1.
Exercise 4: Let f (x) = g(x)h(x) where√g(x) = dx2 e and h(x) = sin 2πx. Check the
continuity of f , g and h at x = 2 and x = 2.
Exercise 5:
(a) Let f, g : R → R be continuous such that given any two points x1 < x2 , there exists
a point x3 such that x1 < x3 < x2 such that f (x3 ) = g(x3 ). Show that f (x) = g(x)
for all x ∈ R.
(b) Let f : R → R be a continuous function such that f (x) = f (x2 ) for all x ∈ R. Show
that f is a constant function.
(c) Suppose f : [0, ∞) → R is continuous and limx→∞ f (x) exists. Show that f is
bounded on [0, ∞).
(d) Let f : [0, 1] → R be continuous such that f (0) = f (1). Show that there exists
x0 ∈ [0, 12 ] such that f (x0 ) = f (x0 + 21 ).
(e) Let f : [0, 2] → R be a continuous function and f (0) = f (2). Prove that there exist
real numbers x1 , x2 in [0, 2] such that x2 − x1 = 1 and f (x2 ) = f (x1 ).
(f) Let I := [a, b] and f : I → R be a continuous function on I such that for each x ∈ I
there exists a y ∈ I such that
1
|f (y)| ≤ |f (x)|.
2
Prove that there exists a point c ∈ I such that f (c) = 0.
(g) Let f : [0, 1] → R is a continuous function such that Range(f ) ⊂ Q. Prove f is a
constant function.
Exercise 6: Find the points of discontinuity of the function f defined by
sin2 (n!πx)
f (x) = lim lim 2
x→∞ t→0 sin (n!πx) + t2
