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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 â†’ a f(x) = l

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

lim x â†’ 1 x 2 and   lim x â†’ 1 3x 2

Sum of the functions

lim x â†’ 1 x 2   +   lim x â†’ 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 â†’ 1 (x 2   + 3x 2 ) =   lim x â†’ 1 4x 2 = 4

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

lim x â†’ 1[f(x) + g(x)] =   lim x â†’ 1f(x) + lim x â†’ 1g(x)

Difference of the functions

lim x â†’ 1 x 2   â€“   lim x â†’ 1 3x 2 = 1 â€“ 3 = â€“ 2

lim x â†’ 1 (x 2   â€“ 3x 2 ) =   lim x â†’ 1 (â€“2x 2 ) = â€“2  lim x â†’ 1 (x 2 ) = â€“2

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

lim x â†’ 1[f(x) â€“ g(x)] =   lim x â†’ 1f(x) â€“ lim x â†’ 1g(x)

Product of two functions
lim x â†’ 1 x 2   ×   lim x â†’ 1 3x 2 = 1 × 3 = 3

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

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

lim x â†’ 1[f(x) â€¢ g(x)] =   lim x â†’ 1f(x) â€¢ lim x â†’ 1g(x)

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

lim x â†’ 1 x 2 lim x â†’ 1 3x 2 = 1 3

lim x â†’ 1 x 2 3x 2 = lim x â†’ 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 â†’ a f(x) g(x) = lim x â†’ a f(x) lim x â†’ a g(x) , with g(x) â‰  0.
Identity
4 ×   lim x â†’ 1 x 2   = 4

lim x â†’ 1 (4x 2 ) = 4 ×    lim x â†’ 1 x 2   = 4
So, for any scalar Î» we have,   lim x â†’ a[Î» .f(x)] = Î» .  lim x â†’ a[f(x)].