Anti-terrorism {*_*}

In a speech to a gathering of mathematics professors from throughout the United States, George W. Bush warned the academics not to misuse their position to force their often extremist political views on young Americans. "It is my understanding", the president said, "that you are frequently teaching algebra classes in which your students learn how to solve equations with the help of radicals. I can't say that I approve of that..."

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Source:

[3]

Root of the problem: Negative infinity

I came across this interesting problem which is quite simple in appearance yet, at the same time, it possesses some qualities of a quiz bee material.

The solution starts with removing the radical by multiplying the expression with its conjugate. From here, there are algebraic routes to the same destination of simplifying the expression which has become a fraction. The denominator now contains the other half of the conjugate.

Here, the gerater trick lies on the concept of absolute value.

The x-squared inside the radical sign can be extracted as x. If x goes to negative infinity, x must be negative. Therefore the root of this term must be negative.

Here is another interesting part. If the true value of x is negative becasue x approaches negative infinity, it is very tempting to over-emphasize the absolute value concept by changing the sign of all x including all the other terms with no root extraction required. Again, one may argue, if x goes to negative infinity as indicated in the limit, then each x in the whole expression is supposed to be covered by that condition. It means x is less than zero or negative. Hence, it is so tempting to rewrite the expression as:

Note that, when simplifying towards the end, x is factored out from the denominator and it cancels with the x in the numerator. As this happens, the rewriting of x outside the radical sign to reflect its true sign becomes somewhat moot.

On the other hand, all x outside the radical sign before extraction, in this case, the terms -2x in the numerator and the first term x in the denominator, are not normally rewritten in texts that I've seen at least, to reflect the true location of x in the number line. There must be some good reason. Before going there, consider:

In this second case, when one changes the sign of all x terms by invoking the blanket effect of absolute value as x goes to negative infinity, the limit is positive 3. On the contrary, when absolute value is invoked only to the roots of extracted term, the limit is negative 3.

It is wise therefore to confine the absolute value effect on root extraction. The other terms outside the radical sign may appear the way they do because they may have acquired their sign from previous mathematical operations. This shouldn't discount that they are essentially negative from the start when the condition says the limit goes to negative infinity. Likewise, when the expression is yet to be simplified, changing the sign of all the other unextracted x can lead the expression to wayward paths during succeeding mathematical operations.

The limit going to positive infinity makes similar problems very easy. On the other hand, when it goes to negative infinity, the concept of absolute value plays a very important role. If not taken into consideration very well, the answer usually becomes either zero or infinity which is wrong.

In the first case, the answer is negative 1.

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