How to study Limits for Engineering Diploma by ednexa.com

1. Limits
Tangent is the limiting position of secant:
Limits
Let f(x) = 3x + 5
As x assumes values which are nearer and nearer to 10, 3x
assumes values which are nearer & nearer to 3 ×10 = 30 and f(x)
i.e. 3x + 5 assumes values which are nearer and nearer to 30 + 5
= 35. In other words, 35 is the limiting value of the function f(x) =
3x + 5 as x tends to 10.
Symbolically,
lim f (x)  35
x 10
Definition
lim
x a
f (x)  
If as x assumes values which are nearer and nearer to a, f(x)
assumes values which are nearer and nearer to ℓ, then ℓ is called
as limiting value of the function f(x) as x tends to a

2. lim
i.e.
x a
f (x)  
Note
By x  2
(x tends to 2)
we mean
1.
x  2 i.e. x  2  0
 division by (x  2) is defined
2.
x assumes values which are nearer & nearer to 2
3.
x approaches 2 either from right or from left of 2

3. Important
1.
a 2  b2   a  b   a  b 
(a  b)  ( a  b) ( a  b)
2. a 3  b3  (a  b) (a 2  ab  b 2 )
3. a 3  b3  (a  b) (a 2  ab  b 2 )
4. (a  b)2  a 2  2ab  b2
5. (a  b)2  a 2  2ab  b2
6. (a  b)3  a 3  3a 2 b  3ab 2  b3
7. (a  b)3  a 3  3a 2 b  3ab2  b3
8.
If value of a polynomial in x is 0 at x = a then
(x-a) is its factor.
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