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Remesh’s Maths Centre

[DIFFERENTIAL CALCULUS]

13
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99,1
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9999, …
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Similarly, if x take values 2
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001,2
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Even then the numerical difference
between x and 2 gets closure to 0
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In general, x  a means that the variable x and x takes values either less than or greater than that
of a and the numerical difference between x and a can be made as small as we please
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It can be symbolically written as: lim f ( x)  k
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x a

xa

(ii) lim  Area of polygon of n sides   Area of circle
nx 

STANDARD RESULTS


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


Limit of a constant function is a constant
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e
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f ( x)  k
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g ( x) = lim f ( x)  lim g ( x)

xa



xa

xa

xa

lim f ( x)
f ( x ) x a
=
, g x   0
lim g ( x)
xa g ( x)

lim

x a



lim n f ( x) = n lim f ( x)

xa

Hsslive
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com
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lim 

E
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:

 x 2  2x  5  2 2  2  2  5 4  4  5 3



3 2  5
65
11
x2  3x  5 
 x 2  3x  2   22  3   2  2 4  6  2 0

iii
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lim 









iv
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g
x a ( x ) 0
xa
xa
lim

The form

0
is called indeterminate form
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

We cannot find the limits such functions directly
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Factorization Method:
a) Factorize the numerator and denominator and cancel the common factors from the numerator and
the denominator
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c) Apply quotient rule of limit
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g
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Substitution Method:
f ( x)
f ( a  h)
= lim

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As x  a , h  0
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sin x sin  0


x   x    0

E
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: Evaluate lim

Hsslive
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com
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Rationalization Method:
a) Rationalize the expression, which involve square roots

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