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Algebra of Limits

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Algebra of Limits - Lesson Summary

 
A limit of a function is written as: lim x → a f(x) = l

Consider f(x) = x 2 and g(x) = 3x 2 such that  lim x → 1 x 2 and   lim x → 1 3x 2 exist.

  lim x → 1 x 2 and   lim x → 1 3x 2
 
Sum of the functions

  lim x → 1 x 2   +   lim x → 1 3x 2 = 1 + 3 = 4

If the functions are first added and then its limit is found as x tends to 1, the value of the function is 4.
 
  lim x → 1 (x 2   + 3x 2 ) =   lim x → 1 4x 2 = 4
 
So, limit of sum of two functions is sum of the limits of the functions i.e.,

  lim x → 1[f(x) + g(x)] =   lim x → 1f(x) + lim x → 1g(x)
 
Difference of the functions
                                                   
  lim x → 1 x 2   –   lim x → 1 3x 2 = 1 – 3 = – 2

   lim x → 1 (x 2   – 3x 2 ) =   lim x → 1 (–2x 2 ) = –2  lim x → 1 (x 2 ) = –2

So, limit of difference of two functions is difference of the limits of the functions i.e., .

  lim x → 1[f(x) – g(x)] =   lim x → 1f(x) – lim x → 1g(x)

Product of two functions
  lim x → 1 x 2   ×   lim x → 1 3x 2 = 1 × 3 = 3

The product of the limits of the functions is equal to 3.
   lim x → 1 (x 2   × 3x 2 ) =   lim x → 1 (3x 4 ) = 3  lim x → 1 (x 4 ) = 3

Limit of the product of the functions is also equal to 3 i.e.,

  lim x → 1[f(x) • g(x)] =   lim x → 1f(x) • lim x → 1g(x)

So, the limit of product of two functions is product of the limits of the functions
Quotient of the functions

lim x → 1 x 2 lim x → 1 3x 2 = 1 3

lim x → 1 x 2 3x 2 = lim x → 1 1 3 = 1 3

So, limit of quotient of two functions is quotient of the limits of the functions with a non-zero denominator, i.e., lim x → a f(x) g(x) = lim x → a f(x) lim x → a g(x) , with g(x) ≠ 0.
Identity
4 ×   lim x → 1 x 2   = 4

  lim x → 1 (4x 2 ) = 4 ×    lim x → 1 x 2   = 4
So, for any scalar λ we have,   lim x → a[λ .f(x)] = λ .  lim x → a[f(x)].

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