Rationalizing the numerator with limits

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The discussion centers on the legality of a mathematical step involving limits and rationalization. The user questions the terminology used, clarifying that the process is actually factoring rather than rationalizing. They demonstrate that substituting values for z and x yields valid results, provided both are nonnegative. The conversation emphasizes the importance of correctly identifying mathematical operations to avoid confusion. Overall, the focus is on understanding the manipulation of expressions in limit problems.
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Homework Statement



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The Attempt at a Solution



I don't see how the above step is legal. I plugged in 4 for z and 9 for z and it works, but what is this move called? it's neither rationalizing the numerator nor rationalizing the denominator.
 
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It's called factoring. z-x=(sqrt(z)-sqrt(x))*(sqrt(z)+sqrt(x)). It should work ok as long as z and x are nonnegative. I think you can assume that because otherwise your numerator isn't very well defined.
 
FOIL the denominator in the second step and you'll get back what you started with.
 

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