Dept. of Mathematics VIT Vellore
CALCULUS and ANALYTICAL GEOMETRY (TMAT103L)
Problem Sheet- Limits
Instructor:S. Balaji
1. For the function f (t) graphed below find the following limits or explain why they
do not exist.
(a) limt→−2 f (t)
(c) limt→0 f (t)
(b) limt→1 f (t)
(d) limt→0.5 f (t)
2. Which of the following statements about the function y = f (x) graphed below are
true and which are false ?.
(a) limx→2 f (x) does not exist.
(b) limx→2 f (x) = 2.
(c) limx→1 f (x) does not exist.
(d) limx→x0 f (x) exist at every point
x0 ∈ (−1, 1).
(e) limx→x0 f (x) exist at every point
x0 ∈ (1, 3).
3. Explain why the following limits do not exist
(a) limx→0
x
|x|
(b) limx→1
1
x−1
(g) limx→1
√ x−1
x+3−2
(h) limx→4
4−x
√
5− x2 +9
4. Find the following
(a) limx→−7 (2x + 5)
3
(b) limh→0 √3h+1+1
1/3
(c) limz→0 (2z − 8)
(d) lims→2/3 3s(2s − 1)
(e) limx→1
(f) limx→0
1
−1
x
x−1
1
1
+ x+1
x−1
x
(i) limx→0 sin2 (x)
√
(j) limx→−π x + 4cos(x + π)
(k) limx→0 (x2 − 1)(2 − cosx)
5. If
√
5 − 2x2 ≤ f (x) ≤
6. If limx→4
f (x)−5
x−2
7. If limx→0
f (x)
x2
5 − x2 for −1 ≤ x ≤ 1, find limx→0 f (x) .
= 1, find limx→4 f (x).
= 1, find
(a) limx→0 f (x)
8. limx→0
√
sinx−tanx
.
x3
1
9. limx→1+ ( lnx
−
1
).
1−x
10. limx→0 xsin( x1 ).
11. limx→π
1+cosx
.
sinx
12. limx→0
ex −e−x
.
x
(b) limx→0
f (x)
x