# Solve. limit as (x, y) approaches (1, 0) for( {square root of {x + y} - square root of {x} )...

## Question:

{eq}\displaystyle \lim_{(x, y) \to (1, 0)} \frac{\sqrt{x + y} - \sqrt{x}}{xy} {/eq}

## Limits of Functions of Several Variables:

Direct substitution into this limit yields the indeterminate form {eq}\displaystyle \frac00 {/eq} and the limit might need to be rewritten. One thing that might be attempted is to try and rationalize the numerator.

## Answer and Explanation:

Consider the argument of the limit.

{eq}\displaystyle \frac{\sqrt{x + y} - \sqrt{x}}{xy} {/eq}

To rationalize the numerator, multiply by the top and bottom by the conjugate {eq}\sqrt{x + y} + \sqrt{x} {/eq}.

{eq}\displaystyle \frac{\sqrt{x + y} - \sqrt{x}}{xy} \cdot \frac{\sqrt{x + y} + \sqrt{x}}{\sqrt{x + y} + \sqrt{x}} = \frac{x+y-x}{ xy (\sqrt{x + y} + \sqrt{x}) } \\ = \displaystyle \frac{y}{ xy (\sqrt{x + y} + \sqrt{x}) } = \frac{1}{ x (\sqrt{x + y} + \sqrt{x}) } \ {/eq}

Plug this into the limit and again try direct substitution.

{eq}\displaystyle \lim_{(x, y) \to (1, 0)} \frac{\sqrt{x + y} - \sqrt{x}}{xy} \\ = \displaystyle \lim_{(x,y) \to (1,0) } \frac{1}{ x (\sqrt{x + y} + \sqrt{x}) } = \frac12 {/eq}