Exercise 13.1
Evaluate the following limits in Exercises 1 to 22.
1. 
Ans. 
3 + 3 = 6
2. 
Ans. 
3. 
Ans. 
4. 
Ans. 
5. 
Ans. 
6. 
Ans. 
Putting 
as 
7. 
Ans. 
= 
= 
8. 
Ans.
=
= 
= 
9. 
Ans. 
10. 
Ans. 
= 
= 
= 
= 
= 1 + 1 = 2
11. 
Ans. 
= 
= 
= 1
12. 
Ans. = 
= 
= 
= 
13. 
Ans.
= 
= 
= 
14. 
Ans. 
= 
= 
= 
= 
15. 
Ans. 
Putting 
as 
= 
= 
= 
= 
= 
16. 
Ans. = 
17. 
Ans. = 
= 
= 
= 
= 
= 
18. 
Ans. 
= 
= 
= 
= 
19. 
Ans. = 
= 
= 
= 0
20. 
Ans. 
Dividing numerator and denominator by 
= 
= 
= 
= 1
21. 
Ans. Given:
= 
= 
= 
= 
= 
= 0
22. 
Ans. Given:
Putting 
as 
= 
= 
= 
= 
= 
23. Find 
and 
where 
Ans. Given:
Now
= 
And
= 3 (1 + 1) = 
24. Find where 
Ans. Given:
L.H.L.
Putting 
as 
= 
= 
= 0
R.H.L.
Putting 
as
= 
= 
= 
25. Evaluate where 
Ans. Given:
L.H.L.
Putting 
as 
= 
= 
= 
R.H.L.
Putting 
as
= 
= 
= 
Here, L.H.L. 
R.H.L.
Therefore, this limit does not exist at 
26. Find where 
Ans. Given:
L.H.L.
Putting as
= 
= 
=
R.H.L.
Putting as
= 
= =
Here, L.H.L. R.H.L.
Therefore, this limit does not exist at 
27. Find 
where 
Ans. Given:
L.H.L.
Putting 
as 
= 
= 
= 0
R.H.L.
Putting 
as
= 
= 
= 0
Here, L.H.L. = R.H.L.
Therefore, this limit exists at and 
28. Suppose 
and if 
what are possible values of 
and 
?
Ans. Given: and
and
……… (i)
Now
Putting as
= 
……….(ii)
Again
Putting as
= 
……….(ii)
Putting values from eq. (ii) and (iii) in eq. (i), we get
and 
On solving these equation, we get
and 
29. Let 
be fixed real numbers and define a function 
What is 
? For some 
compute 
Ans. Given:
Now
= 
= 
= 0
Also
= 
30. If 
for what values of does 
exists?
Ans. Given: 
exists for all 
Now L.H.L.
Putting as
= 
= 
= 0 + 1 = 1
Also R.H.L.
Putting as
= 
= 
= 
Here, L.H.L. R.H.L.
Therefore, this limit does not exist.
31. If the function 
satisfies 
valuate 
Ans. Since
32. If 
for what integer 
and 
does both and exist?
Ans. Both and exist.
and
Now
= 
= 
And
= 
= 
……….(i)
Therefore, for exists we need 
Again
= 
= 
And
= 
= 
Therefore, 
exists for any integral value of and 
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