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