Question

How do you find `lim sqrt(3x+1)-2sqrtx` as `x->oo`?

Answer 1

`lim_(x->oo)(sqrt(3x+1)-2sqrt(x))=-oo`

Explanation:

The question asks to find `lim_(x->oo)(sqrt(3x+1)-2sqrt(x))`.

Multiply by `(sqrt(3x+1)+2sqrt(x))/(sqrt(3x+1)+2sqrt(x))` to get
`=lim_(x->oo)((sqrt(3x+1)-2sqrt(x))*(sqrt(3x+1)+2sqrt(x))/(sqrt(3x+1)+2sqrt(x)))`
`=lim_(x->oo)((3x+1-4x)/(sqrt(3x+1)+2sqrt(x)))`
`=lim_(x->oo)((1-x)/(sqrt(3x+1)+2sqrt(x)))`
`=lim_(x->oo)((1-x)/(sqrt(x)(sqrt(3+1/x)+2)))`
`=lim_(x->oo)((1/sqrt(x)-sqrt(x))/(sqrt(3+1/x)+2))`
`=(lim_(x->oo)(1/sqrt(x)-sqrt(x)))/(lim_(x->oo)(sqrt(3+1/x)+2))`

The numerator evaluates to `-oo` while the denominator is a finite number.

Thus, the answer is `-oo`.

This can be seen clearly with a graph:
graph{sqrt(3x+1)-2sqrt(x) [-1, 100, -5, 2]}