The document discusses limits, continuity, and differentiability. It defines the limit of a function, continuity of a function at a point using three conditions, and Cauchy's definition of continuity using delta and epsilon. It also discusses left and right continuity, Heine's definition of continuity using convergent sequences, and the formal definition of continuity. Examples are provided to illustrate calculating limits and determining continuity.

1. Limit, Continuity and Differentiability
Limits
1.
 e x  e sin x 
lim 

x  0  x  sin x  is equal to


(a) –1
(b) 0
(c) 1
(d) None of these

2. Ans:
(c) 1
 e x  e sin x
lim 
x 0  x  sin x



,

0

 form 
0

Using L-Hospital’s rule three times, then
e x  e sin x . cos x
e x  e sin x cos 2 x  sin x .e sin x
lim
 lim
x 0
x 0
1  cos x
sin x
e x  e sin x . cos 3 x  e sin x 2 cos x sin x  e sin x . cos x sin x  e sin x . cos x
 lim
x 0
cos x
= 1.

3. 2. The value of the constant α and β such that
 x2 1

lim 
 x     0

x   x  1


are respectively
(a) (1, 1)
(b) (–1, 1)
(c) (1, –1)
(d) (0, 1)

4. Ans:
(c) (1, –1)
 x2 1


lim 
 2x     0

x  x  1


x 2 (1   )  x (   )  1  b
lim
0
x 
x 1
Since the limit of the given expression is
zero, therefore degree of the polynomial in
numerator must be less than denominator.
 1 – α = 0 and α + β = 0 , α = 1 and β = 1.

5. 3. Let f: RR be a differentiable function
f (x )
 1 
4 t3
f (2)  6, f ' (2)   .
having
Then xlim2  x  2 dt

 48 
6
equals
(a) 12
(b) 18
(c) 24
(d) 36

6. Ans:
(b) 18
f (x )

lim
x 2
6
4 t 3 dt
x 2
4( f (x ))3  f ' (x )
(0 / 0 form)  lim
x 2
1
= 4 (f(2))3 × f(2) = 18.

7. 1  n2
lim
4. The value of n    n will be
(a) – 2
(b) – 1
(c) 2
(d) 1

8. Ans:
(a) – 2
1  n 2  lim (1  n)(1  n)
2 (1  n)
lim
n 1
 lim
n(n  1)
n n
n
n
2
1

 lim 2   1 
n  n
 =2(0 – 1) = -2

9. 1  2  3  ....n
5. The value of n  n 2  100
is equal
lim
(a) ∞
(b) 1/2
(c) 2
(d) 0

10. Ans:
1  2  3  .....  n
We have, n
n 2  100
lim
1

n 2 1  
n
n(n  1)
1

 lim
 lim

n  2(n 2  100 )
n 
100  2 .

2n 2  1  2 
n 


11. Continuity
Introduction
The word ‘Continuous’ means without any
break or gap. If the graph of a function has no
break, or gap or jump, then it is said to be
continuous.
A function which is not continuous is called a
discontinuous function.
While studying graphs of functions, we see that
graphs of functions sin x, x, cos x, ex etc. are
continuous but greatest integer function [x] has
break at every integral point, so it is not
continuous.

12. 1
Similarly tan x, cot x, sec x,
etc. are also
x
discontinuous function.

13. Continuity of a Function at a Point
A function f(x) is said to be continuous at a
point x = a of its domain iff.
lim f (x)  f (a)
xa
i.e. a function f(x) is continuous at x = a if and
only if it satisfies the following three conditions :
1. f(a) exists. (‘a’ lies in the domain of f)
2. lim f (x) exist i.e. lim f (x)  lim f (x) or
xa
R.H.S. = L.H.S.
xa
xa

14. 3. lim f (x)  f (a)
xa
function).
(limit equals the value of

15. Cauchy’s definition of continuity
A function f is said to be continuous at a point a
of its domain D if for every ∊ > 0 there exists δ
> 0 (dependent on ∊) such that
| x  a |  |f (x)  f (a) | .
Comparing this definition with the definition of
limit we find that f(x) is continuous at x = a if
lim f (x) exists and is equal to f(a) i.e., if
xa
lim f (x)  f (a)  lim f (x) .
xa
xa 

16. Heine’s definition of continuity
A function f is said to be continuous at a point a
of its domain D, converging to a, the sequence
<an> of the points in D converging to a, the
sequence <f(an)> converges to f(a) i.e. lim an =
a ⇒ lim f(an) = f(a).
This definition is mainly used to prove the
discontinuity to a function.
Continuity of a function at a point, we find its
limit and value at that point, if these two exist
and are equal, then function is continuous at
that point.

17. Formal definition of continuity
The function f(x) is said to be continuous at x =
a in
its domain if for any arbitrary chosen positive
number
∊ > 0, we can find a corresponding number δ
depending on ∊ such that |f(x) – f(a)| < ∊ ∀ x
for
which 0 < | x – a| < δ.
Continuity from Left and Right
Function f(x) is said to be
1. Left continuous at x = a if
lim f (x)  f (a)
xa0
2. Right continuous at x = a if
Lim f (x)  f (a) .
xa 0

18. Thus a function f(x) is continuous at a point x =
a
if it is left continuous as well as right continuous
at
x = a.
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