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: