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Theorem on Limits:
Let  and  . If l and m exist then,
1)  .
2)  .
3)  Provided m ¹ 0.
4)  Where k is constant.
5) If f(x) £ g(x) then l £ m.
6)  .
7) If f(x) £ g(x) £ h(x) for all x.
and  then  (Squeeze play/ Sandwitch Theorem).
Illustration 3:
If [x] denotes the integral part of x, then find  .
Solution:
Let 
Now we know
\ Adding them all gives us
By using squeeze play theorem we get,
 .
Some important expansions (Power Series):
1)  .
2)  Here -1 < x £ 1.
3)  .
4)  .
5)  .
6)  .
7)  .
8) 
9)  (For rational or integral n).
Illustration 4:
Find the series expansion of Sin2x?
Solution:
Now we know that
Note that many other series could be found in that way as we found the series for Sin2x.
Some Standard results on limits:
1) 
2) 
3) 
4) 
5) 
6) 
7) 
8) 
9) 
10) 
Note: These limits could be derived using the series expansion or by L1 Hospital’s rule which will discussed in a later section.
Illustration 5:
Find the value of  ?
Solution:
Evaluation of Limits:
1) Direct Substitution:
If we get a finite number by direct substitution of point we are done.
Illustration 6: Find  ?
Solution:
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Vriti Edustore
