If limx-->a f(x)= ∞ and limx-->g(x)=c, where c is a real number. Prove a)limx-->[f(x)+g(x)]= ∞ b)limx-->a[f(x)g(x)]= ∞ if c>0

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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Here we need to memorize some of the properties of algebraic limit theorem.

`lim_(xrarra)[f(x)+g(x)] = lim_(xrarra) f(x)+lim_(xrarra) g(x)`

`lim_(xrarra)[f(x)*g(x)] = lim_(xrarra) f(x)*lim_(xrarra) g(x)`


So if we apply this in to our question;

`lim_(xrarra)[f(x)+g(x)] `

`= lim_(xrarra) f(x)+lim_(xrarra) g(x)`

`= oo+c`

`= oo`


`lim_(xrarra)[f(x)*g(x)] `

`= lim_(xrarra) f(x)*lim_(xrarra) g(x)`

`= oo*c`

= `oo`     {since c>0. If c<0 it will become `-oo` }


oo is the number without limit. So it is the largest possible number we have.

So if we add c to infinity; `c+oo = oo`

Even we multiply c by infinity; `c*oo = oo`


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