Summary Mind map

LIMITS
Some Useful Expansio
1. Right hand and Left hand Limit (i) sin x = x –
x3
+
x
3! 5
x2 x
To evaluate lim f x  (ii) cos x = 1 –
2!
+
4
x  a
3
2
1. Put x = a + h in f (x) to get lim f a  h   (iii) tan x = x + x +
h0 3 15
(iv) sin–1 x = x
2. Take the limit as h → 0.
12. x 3 12. 32 5
To evaluate lim f x .  +
3!
+
5!
X +
x  a
1. Put x = a – h in f (x) to get lim f a  h .
h0
  (v) (sin–1 x)²
2. Take the limit as h → 0.
2 2.22 4
= x² + x
2! 4!

2. Some Useful Limits (vi) tan–1 x = x –
x3
3
+

(4) Exponential and Logar
x n  an
(1) lim = nan – 1, where n ∈ Q, the set of rational numbers.
x a x a ex  1
(i) lim =1
x 0 x
(2)
ax  1
 (ii) lim = loge a
x 0 x

 , if a  1 ax  bx
n  (iii) lim = loge
(i) lim a =  1, if a  1 x 0 x
n 
 0, if  1  a  1

1  x 
n
doesnot exist , if a  1 1
 (iv) lim =n
x 0 x
a0 x p  a1 x p  1  ap  1 x  ap n
(ii) lim  1
x  q
b0 x  b1 x q1
 bq  1 x  bq (v) lim  1 +  = e
n→0  n

 
1/h
a0 (vi) lim 1  ah =ea
 , if p  q h0
 b0

 log x