I've been going through the diagnostic tests for my Calculus Textbook to get ready for classes starting on Monday. One of the questions is this:
Rationalize the expression and simplify.
$\frac{\sqrt{4+h}-2}{h}$
The answer is $\frac{1}{\sqrt{4+h}+2}$.
I understand how to get to that equation, but I don't understand why. Wasn't the denominator of the equation already rational? What's the point of rationalizing the numerator in this case?
Best Answer
I don't like the term rationalize here. What you've really done is put the expression in a form where plugging in zero for $h$ makes sense: you can't do this in the first form because you can't divide by zero. That is likely what the textbook author's intent was. I think they created this as a silly pre-exercise to what you'll be doing later so you'll "be prepared" to evaluate limits. Soon you'll do something that looks like $$ \lim_{h \to 0}\frac{\sqrt{4+h}-2}{h} = \lim_{h \to 0}\frac{1}{\sqrt{4+h}+2} = \frac{1}{\sqrt{4+0}+2} = \frac{1}{4} $$