Question

How do you solve the following limit `(sqrt(x+3)-sqrt(x))/(sqrt(x+2)-sqrt(x+1))` as x approaches infinity?

Answer 1

`3`

Explanation:

In the first step we multiply numerator and denominator bys

`sqrt(x+3)+sqrt(x)`
and then by
`sqrt(x+2)+sqrt(x+1)`
and we get

`3(sqrt(x+2)+sqrt(x+1))/(sqrt(x+3)+sqrt(x))`
and then leave aside `sqrt(x)` in the numerator and denominator

`(3sqrt(x)(sqrt(1+2/x)+sqrt(1+1/x)))/(sqrt(x)(sqrt(1+3/x)+1))`
cancelling the square root term we get

`6/2=3` as the searcherd Limit if `x`tends to infinity.