Question

Use the properties of limits to find the limit.
Picture provided below

Answer 1

`(4x)/(x-5)=\frac4{1-\frac5x}`, and as
`x\to\infty`, the
`\frac5x` term vanishes.

`(4x)/(x^2+5)=\frac{\frac4x}{1+\frac5{x^2}}`, and as
`x\to\infty`, both
`\frac4x` and
`\frac5{x^2}` vanish.

So

`\displaystyle\lim_(x\to\infty)\left((4x)/(x-5)+(4x)/(x^2+5)\right)=\lim_(x\to\infty)(4+0)=4`

and the answer is C.

Answer 2

Option C. 4 is the correct option.

Explanation:

The given expression is
`\lim_(x\rightarrow \oe )[(4x)/(x-5)+(4x)/(x^(2)+5)]`

We have to calculate the limit of the given expression.

Now we will convert the equation in the form as given below

`=\lim_(x\rightarrow \oe )[(4)/(1-(5)/(x))+(4)/(x+(5)/(x))]`

We have done this transformation of the equation because we know

`\lim_(x\rightarrow \oe )[(1)/(x)]=0`

Now by the property of limit second term in the question will be vanished

and the remaining part will be

`\lim_(x\rightarrow \oe )[(4)/(1-(5)/(x))]`

Now by putting x→∞ we get

`[(4)/(1-0)]=4`

Therefore option C is the correct answer.