Solution assignment 04 The basics

Assignment 4

Calculate the value of the following fraction as $h$ approaches to $0$ :

$\displaystyle\frac{\sqrt{x+h}-\sqrt{x}}{h}$

Solution

If we substitute $h=0$ we get an undefined result. Therefore we use a trick: we multiply both numerator and denominator by:

$\sqrt{x+h}+\sqrt{x}$

and get:

$\displaystyle\frac{(\sqrt{x+h}+\sqrt{x})(\sqrt{x+h}-\sqrt{x})}{h(\sqrt{x+h}+\sqrt{x})}$

We rewrite the numerator by applying one of the special products and finally we can safely substitute $h=0$ :

$\displaystyle\frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})}=\displaystyle\frac{h}{h(\sqrt{x+h}+\sqrt{x})}=\displaystyle\frac{1}{\sqrt{x+h}+\sqrt{x}}=\displaystyle\frac{1}{2\sqrt{x}}$

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Solution assignment 04 The basics

Assignment 4

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