Math CentralQuandaries & Queries


Question from autumn, a student:

given: f(x)=x/(x^2+2) determine: [f(x-h)-f(x)]/h

Hi Autumn,

To evaluate $f(x - h)$ look at the expression for $f(x)$ and replace $x$ everywhere you see it by $x - h.$ $f(x - h)$ then becomes

\[\frac{x - h}{\left((x - h)^2 - f(x)\right)}.\]

Expand the denominator and then write down $f(x - h) - f(x).$ This is the difference of two fractions so you need to find a common denominator and rewrite $f(x - h) - f(x)$ with this denominator. Simplify. If you do this correctly you will see that the numerator of the rewritten $f(x - h) - f(x)$ has a common factor of $h$ which will cancel with the denominator of

\[\frac{f(x - h) - f(x)}{h}\]

as long as $h \neq 0.$


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