Limits & Continuity Worksheet Class 11 Notes | EduRev

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Class 11 : Limits & Continuity Worksheet Class 11 Notes | EduRev

 Page 1


WORKSHEET II
Course: Mat101E
Topic: Limits and Continuity
1. Use ?-d denition to prove that:
(a) lim
x?-4
v
1-2x = 3
(b) lim
x?
v
3
1
x
2
=
1
3
(c) lim
x?2
x
2
-1
x+3
=
3
5
(d) lim
x?1
1
v
x+1
=
1
2
(e) lim
x?2
(x-2)
3
sin
1
x-2
= 0
(f) lim
x?1
(x
2
-1)cos
1
x-1
= 0
(g) lim
x?1
(x
2
+x+3) = 5
(h) lim
x?1
(x
2
+x+3)?= 1
2. For the following limits, nd the appropriate value of d that corresponds to the given ? value:
(a) lim
x?-1
(1-2x) = 3, ? = 0.01
(b) lim
x?0
1
x-1
=-1, ? = 0.5
(c) lim
x?2
v
11-x = 3, ? = 1
3. Find the following limits, if they exist, or explain why they do not exist.
(a) lim
x?64
v
x-8
3
v
x-4
(b) lim
x?1
1-
v
x
1-x
(c) lim
x?27
v
x-3
v
3
3
v
x-3
(d) lim
x?0
3
v
x+1-1
4
v
x+1-1
(e) lim
x?0
x
tan3x
(f) lim
x?
sinx
x-p
(g) lim
x?0
sin
2
x
x(1-cosx)
(h) lim
x?8
x+sinx
x+cosx
(i) lim
x?±8
tan
-1
x
(j) lim
x?8
v
x+x
v
x+cosx
(k) lim
x?8
e
x
sin(e
-x
)
(l) lim
x?8
x
2
+2
x-5
(m) lim
x?8
x-5
x
2
+2
(n) lim
h?0
cosh-1
h
(o) lim
x?8
2x
sinx
(p) lim
x?8
x+sinx
2x+5
(q) lim
x?0
tan
-1
x
x
(r) lim
x?0
1
2+x
-
1
2
x
(s) lim
x?0
xsin
1
x
(t) lim
x?0
sin
2
3x
5x
2
(u) lim
x?0
v
1+tanx-
v
1+sinx
x
3
(v) lim
x?0
v
1+x
2
-
v
1+x
x
(w) lim
x?0
1+sinx-cosx
1-sinx-cosx
(x) lim
x?8
x(
v
9x
2
+1-3x)
(y) lim
x?3
3-x
v
4-x-
v
x
3
(z) lim
x?8
(
v
x
2
+1-
v
x
2
-1)
1
Page 2


WORKSHEET II
Course: Mat101E
Topic: Limits and Continuity
1. Use ?-d denition to prove that:
(a) lim
x?-4
v
1-2x = 3
(b) lim
x?
v
3
1
x
2
=
1
3
(c) lim
x?2
x
2
-1
x+3
=
3
5
(d) lim
x?1
1
v
x+1
=
1
2
(e) lim
x?2
(x-2)
3
sin
1
x-2
= 0
(f) lim
x?1
(x
2
-1)cos
1
x-1
= 0
(g) lim
x?1
(x
2
+x+3) = 5
(h) lim
x?1
(x
2
+x+3)?= 1
2. For the following limits, nd the appropriate value of d that corresponds to the given ? value:
(a) lim
x?-1
(1-2x) = 3, ? = 0.01
(b) lim
x?0
1
x-1
=-1, ? = 0.5
(c) lim
x?2
v
11-x = 3, ? = 1
3. Find the following limits, if they exist, or explain why they do not exist.
(a) lim
x?64
v
x-8
3
v
x-4
(b) lim
x?1
1-
v
x
1-x
(c) lim
x?27
v
x-3
v
3
3
v
x-3
(d) lim
x?0
3
v
x+1-1
4
v
x+1-1
(e) lim
x?0
x
tan3x
(f) lim
x?
sinx
x-p
(g) lim
x?0
sin
2
x
x(1-cosx)
(h) lim
x?8
x+sinx
x+cosx
(i) lim
x?±8
tan
-1
x
(j) lim
x?8
v
x+x
v
x+cosx
(k) lim
x?8
e
x
sin(e
-x
)
(l) lim
x?8
x
2
+2
x-5
(m) lim
x?8
x-5
x
2
+2
(n) lim
h?0
cosh-1
h
(o) lim
x?8
2x
sinx
(p) lim
x?8
x+sinx
2x+5
(q) lim
x?0
tan
-1
x
x
(r) lim
x?0
1
2+x
-
1
2
x
(s) lim
x?0
xsin
1
x
(t) lim
x?0
sin
2
3x
5x
2
(u) lim
x?0
v
1+tanx-
v
1+sinx
x
3
(v) lim
x?0
v
1+x
2
-
v
1+x
x
(w) lim
x?0
1+sinx-cosx
1-sinx-cosx
(x) lim
x?8
x(
v
9x
2
+1-3x)
(y) lim
x?3
3-x
v
4-x-
v
x
3
(z) lim
x?8
(
v
x
2
+1-
v
x
2
-1)
1
4. Let lim
x?1
f(x) =-1. Evaluate lim
x?1
sin(1+f(x))
1-f
2
(x)
.
5. Find the right-hand and the left-hand limits of the following functions at the given point(s).
(a) y =
|x-1|
x-1
+x
2
, (x = 1)
(b) y =
1
3-4
1
x2
,(x = 2)
(c) y =
1
x
4=3
-
1
(x-2)
1=3
, (x = 0,2)
(d) y =
2+x
1+2
1=x
, (x = 0)
(e) y =
v
1-cos2x
v
2x
, (x = 0)
(f) y = tan
-1
x
x-2
,(x = 2)
(g) y =
x
x
2
-1
+x
2
, (x =±1)
(h) y =
?
?
?
1-x
2
, |x|= 1
1
|x|
, |x| > 1
, (x =-1)
6. Suppose that f is an even function of x. Does knowing that lim
x?2

f(x) = 7 tell you anything
about either lim
x?-2

f(x) or lim
x?-2
+
f(x)? Give reasons for your answer.
7. Find the asymptotes, if any, of the following functions.
(a) f(x) =
x
3
4-x
2
(b) g(x) =
x
3
+2x-1
x
3
+2x
2
-x-2
8. (a) Graph the following function f.
(b) Find the points, if any, at which f is discontinuous and classify their types.
f(x) =
?
?
?
?
?
?
?
?
?
?
?
?
?
x+3, -3= x <-1
-1, x =-1
-x+1, -1 < x= 1
1
x-1
, 1 < x= 2
x, x > 2
9. Discuss the limit, one-sided limit, continuity and one-sided continuity of f and g at each of the
points x = 0,±1.
(a) f(x) =
?
?
?
?
?
?
?
?
?
?
?
1, x=-1
-x, -1 < x < 0
1, x = 0
-x, 0 < x < 1
1, x > 1
(b) g(x) =
?
?
?
?
?
?
?
0, x=-1
1/x, |x| < 1
0, x = 1
1, x > 1
10. For the following functions, nd the discontinuity points, if any, and classify the types of the
discontinuities.
(a) f(x) =
x-2
x+2
(b) f(x) =
x
2
+1
x
2
-4x+3
(c) f(x) =
1
x
2
+1
(d) f(x) =
|x|
x
2
Page 3


WORKSHEET II
Course: Mat101E
Topic: Limits and Continuity
1. Use ?-d denition to prove that:
(a) lim
x?-4
v
1-2x = 3
(b) lim
x?
v
3
1
x
2
=
1
3
(c) lim
x?2
x
2
-1
x+3
=
3
5
(d) lim
x?1
1
v
x+1
=
1
2
(e) lim
x?2
(x-2)
3
sin
1
x-2
= 0
(f) lim
x?1
(x
2
-1)cos
1
x-1
= 0
(g) lim
x?1
(x
2
+x+3) = 5
(h) lim
x?1
(x
2
+x+3)?= 1
2. For the following limits, nd the appropriate value of d that corresponds to the given ? value:
(a) lim
x?-1
(1-2x) = 3, ? = 0.01
(b) lim
x?0
1
x-1
=-1, ? = 0.5
(c) lim
x?2
v
11-x = 3, ? = 1
3. Find the following limits, if they exist, or explain why they do not exist.
(a) lim
x?64
v
x-8
3
v
x-4
(b) lim
x?1
1-
v
x
1-x
(c) lim
x?27
v
x-3
v
3
3
v
x-3
(d) lim
x?0
3
v
x+1-1
4
v
x+1-1
(e) lim
x?0
x
tan3x
(f) lim
x?
sinx
x-p
(g) lim
x?0
sin
2
x
x(1-cosx)
(h) lim
x?8
x+sinx
x+cosx
(i) lim
x?±8
tan
-1
x
(j) lim
x?8
v
x+x
v
x+cosx
(k) lim
x?8
e
x
sin(e
-x
)
(l) lim
x?8
x
2
+2
x-5
(m) lim
x?8
x-5
x
2
+2
(n) lim
h?0
cosh-1
h
(o) lim
x?8
2x
sinx
(p) lim
x?8
x+sinx
2x+5
(q) lim
x?0
tan
-1
x
x
(r) lim
x?0
1
2+x
-
1
2
x
(s) lim
x?0
xsin
1
x
(t) lim
x?0
sin
2
3x
5x
2
(u) lim
x?0
v
1+tanx-
v
1+sinx
x
3
(v) lim
x?0
v
1+x
2
-
v
1+x
x
(w) lim
x?0
1+sinx-cosx
1-sinx-cosx
(x) lim
x?8
x(
v
9x
2
+1-3x)
(y) lim
x?3
3-x
v
4-x-
v
x
3
(z) lim
x?8
(
v
x
2
+1-
v
x
2
-1)
1
4. Let lim
x?1
f(x) =-1. Evaluate lim
x?1
sin(1+f(x))
1-f
2
(x)
.
5. Find the right-hand and the left-hand limits of the following functions at the given point(s).
(a) y =
|x-1|
x-1
+x
2
, (x = 1)
(b) y =
1
3-4
1
x2
,(x = 2)
(c) y =
1
x
4=3
-
1
(x-2)
1=3
, (x = 0,2)
(d) y =
2+x
1+2
1=x
, (x = 0)
(e) y =
v
1-cos2x
v
2x
, (x = 0)
(f) y = tan
-1
x
x-2
,(x = 2)
(g) y =
x
x
2
-1
+x
2
, (x =±1)
(h) y =
?
?
?
1-x
2
, |x|= 1
1
|x|
, |x| > 1
, (x =-1)
6. Suppose that f is an even function of x. Does knowing that lim
x?2

f(x) = 7 tell you anything
about either lim
x?-2

f(x) or lim
x?-2
+
f(x)? Give reasons for your answer.
7. Find the asymptotes, if any, of the following functions.
(a) f(x) =
x
3
4-x
2
(b) g(x) =
x
3
+2x-1
x
3
+2x
2
-x-2
8. (a) Graph the following function f.
(b) Find the points, if any, at which f is discontinuous and classify their types.
f(x) =
?
?
?
?
?
?
?
?
?
?
?
?
?
x+3, -3= x <-1
-1, x =-1
-x+1, -1 < x= 1
1
x-1
, 1 < x= 2
x, x > 2
9. Discuss the limit, one-sided limit, continuity and one-sided continuity of f and g at each of the
points x = 0,±1.
(a) f(x) =
?
?
?
?
?
?
?
?
?
?
?
1, x=-1
-x, -1 < x < 0
1, x = 0
-x, 0 < x < 1
1, x > 1
(b) g(x) =
?
?
?
?
?
?
?
0, x=-1
1/x, |x| < 1
0, x = 1
1, x > 1
10. For the following functions, nd the discontinuity points, if any, and classify the types of the
discontinuities.
(a) f(x) =
x-2
x+2
(b) f(x) =
x
2
+1
x
2
-4x+3
(c) f(x) =
1
x
2
+1
(d) f(x) =
|x|
x
2
(e) f(x) =
1
1-3
3x
x
(f) f(x) =
{
1-cosx
x
2
, x?= 0
1, x = 0
(g) f(x) =
v
xsin
1
x
(h) f(x) =
?
?
?
?
?
?
?
sin
-1
x
2
, 0 < x < 2
p, x = 2
tan
-1
1
x-2
, x > 2
11. Evaluate the following limit (Do not use the L'H^ opital's Rule).
lim
x?1
+
{ln[sin(x
2
-1)]-ln(x-1)}
12. Dene f(1) in a way that extends f(x) =
x
2
+2x-3
x
2
-1
to be continuous at x = 1.
13. For what value of a, is f(x) =
{
x
2
-1, x < 3
2ax, x= 3
continuous at every x?R?
14. If x
4
= f(x)= x
2
for all x? [-1,1], and x
2
= f(x)= x
4
for all x? (-8,-1)?(1,8), then
at which point(s) c do you automatically know lim
x?c
f(x)? What is the value of the limit at this
point(s)?
15. Show that the equation x
3
-2x+2 = 0 must have a solution between-2 and 0.
16. Show that the following functions have at least one real root.
(a) f(x) =
3
v
x+x-2 (b) g(x) = cosx+sinx-x
17. Suppose that f is a continuous function on the closed interval [0,1] and that 0= f(x)= 1 for
every x? [0,1]. Show that there must exist a number c? [0,1] such that f(c) = c.
18. Suppose that f and g are continuous functions on [a,b], and that f(a) < g(a) and f(b) > g(b).
Prove that f(c) = g(c) for some c? [a,b].
19. If F(x) = (x-a)
2
(x-b)
2
+x where a,b? R. Show that there must exist a number c? (a,b)
such that F(c) =
a+b
2
.
3
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