Indian Institute of Technology Ropar
Department of Mathematics
MA101 - Calculus
First Semester of Academic Year 2022-23
Tutorial Sheet - 1 Continuity, Differentiability and MVT Date: November 10, 2022
1. Evaluate the limits of the following, if they exist:
(a) lim
x→−2
1
x−1
2
x3−8(b) lim
x→1
xm−1
xn−1, m, n ∈N(c) lim
x→0−|x|
xand lim
x→0+|x|
x
(d) lim
x→0x[x] (e) lim
x→1x2[x] (f) lim
x→2(−1)[x]−[x2], where [x] is greatest integer ≤x.
2. Let f:R−→ Rbe defined by
f(x) = 1 if x∈Q
0 if x∈Qc,
where Qdenote the set of rational numbers. Using −δdefinition of limit, prove that limit of
f(x) does not exist at any point in R.
3. Discuss the continuity of the following functions:
(a) f(x) = sin 1
xwhen x6= 0 and f(0) = 0.
(b) f(x) =
x
[x]; if 1 ≤x < 2
1; if x= 2
√6−x; if 2 < x ≤3
4. Using −δdefinition, prove the continuity of the following functions:
(a) f(x) = cos x, for all x∈R.
(b) f(x) = x2cos 1
x, if x6= 0 and f(0) = 0.
(c) f(x) = xsin 1
x, if x6= 0 and f(0) = 0.
5. Determine the points of discontinuity of the function:
(a) g(x) = 1
2−4 cos(3x)(b) h(x) = ex2+1
ex−2e1−x.
6. Prove that the function f(x) =
xrsin 1
x, x 6= 0
0, x = 0
is
(a) continuous from the right at 0 ⇔r > 0.
(b) differentiable from the right at 0 ⇔r > 1.
(c) differentiable at 0 but f0is not continuous at 0, when r= 2.
7. Find the domain of continuity and differentiability of the following functions and find its deriva-
tive.
(a) f(x) = |sin x|(b) g(x) = x−[x] (c) h(x)=2x+|x+ 1|.
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