Given `lim_(x->4) f(x)=7` and `lim_(x->4) g(x)=5` , evaluate: `lim_(x->4)[f(x)+g(x)/5f(x)]`

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tiburtius | High School Teacher | (Level 2) Educator

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We will use the following three theorems:

Theorem 1

Let `L=lim_(x->a)f(x)` and `M=lim_(x->a)g(x)` for some real numbers `L` and `M`.

`lim_(x->a)[f(x)+g(x)]=lim_(x->a)f(x)+lim_(x->a)g(x)=L+M`

Theorem 2

Let  `L=lim_(x->a)f(x)` and  `M=lim_(x->a)g(x)` for some real numbers `L` 
and `M`.

`lim_(x->a)[f(x)g(x)]=lim_(x->a)f(x)cdot lim_(x->a)g(x)=LM`

Theorem 3

Let  `L=lim_(x->a)f(x)` and  `M=lim_(x->a)g(x) ne 0`.

`lim_(x->a)(f(x))/(g(x))=(lim_(x->a)f(x))/(lim_(x->a)g(x))=L/M`

When we use those theorems we get:

`lim_(x->4)[f(x)+(g(x))/(5f(x))]=lim_(x->4)f(x)+(lim_(x->4)g(x))/(5lim_(x->4)f(x))=7+5/(5cdot7)=7+1/7=50/7`

 

 

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