I'm reviewing Algebra in an attempt to review Calculus and came upon a question in the Algebra Diagnostic that asked to rationalize an expression and simplify.
To my understanding, rationalization is the process of rewriting a given expression so that the denominator is non-zero. The equation is, thus:
$$\frac{\sqrt{4+h} - 2}{h}$$
I understand that $h$ can equal zero, and therefore cannot be in the denominator. The answer given is:
$$\frac{1}{\sqrt{4+h} + 2}$$
The denominator in the answer, I believe, is the conjugate of the numerator in the question, but how did they arrive at this state? And in general, is my understanding of rationalization flawed? What is the point and how will that be applied at higher levels of math?
Thank you
The expression $\dfrac{\sqrt{4+h} -2} h$ is undefined when $h=0,$ since then both the numerator and the denominator are equal to $0.\vphantom{\dfrac11}$
The expression $\dfrac 1 {\sqrt{4+h} + 2}$ is equal to the expression above whenever $h\ne0,$ but is defined when $h=0,$ and is continuous where $h=0,$ and when $h=0$ it is equal to $\dfrac 1 4.$
One would rationalize the numerator in the first fraction above for the purpose of showing that it approaches $\dfrac 1 4$ as $h$ approaches $0.$
If the denominator of a fraction approaches $0$ while the numerator approaches some nonzero number, then the fraction approaches $\infty,$ in absolute value at least. But if both the numerator and the denominator approach $0,$ then in many cases the fraction approaches some particular number that is not $0$ or $\pm\infty.$ That is important in calculus since derivatives are limits as both the numerator and the denominator of a difference quotient approach $0.$