\(\displaystyle \lim_{x \to \infty} \sqrt{x^2 + a \cdot x} \;-\; \sqrt{x^2 + b \cdot x}\)the parameters become insignificant as x becomes infinite.-- If we apply that reason here - the limit becomes 0.
I tried to get the variable into the denominator of fractions (just as you did) on one of my first attempts by multiplying the original expression by sqrt(x^2)/sqrt(x^2). But, of course, that led to sqrt(1) - sqrt(1).
This type of sloppiness has bitten me before -- in that I do not always go back to the beginning to verify a result when I make assumptions. I hope I remember your lesson ...
Thank you for your confirmation. I worked this exercise from a few different approaches, so that by the time I arrived at my posted result I was confident that the algebra is sound.
I simply did not arrive at the result I wanted, which is the form that PKA posted.
Can you identify the flaw in the following?
\(\displaystyle \sqrt{x^2 + a \cdot x} \;-\; \sqrt{x^2 + b \cdot x}\)
\(\displaystyle \frac{\sqrt{x^2}}{\sqrt{x^2}} \cdot \left(\sqrt{x^2 + a \cdot x} \;-\; \sqrt{x^2 + b \cdot x}\right)\)
\(\displaystyle \sqrt{x^2} \cdot \left(\frac{\sqrt{x^2 + a \cdot x}}{\sqrt{x^2}} \;-\; \frac{\sqrt{x^2 + b \cdot x}}{\sqrt{x^2}}\right)\)
Thank you for your confirmation. I worked this exercise from a few different approaches, so that by the time I arrived at my posted result I was confident that the algebra is sound.
I simply did not arrive at the result I wanted, which is the form that PKA posted.
Can you identify the flaw in the following?
\(\displaystyle \sqrt{x^2 + a \cdot x} \;-\; \sqrt{x^2 + b \cdot x}\)
\(\displaystyle \frac{\sqrt{x^2}}{\sqrt{x^2}} \cdot \left(\sqrt{x^2 + a \cdot x} \;-\; \sqrt{x^2 + b \cdot x}\right)\)
\(\displaystyle \sqrt{x^2} \cdot \left(\frac{\sqrt{x^2 + a \cdot x}}{\sqrt{x^2}} \;-\; \frac{\sqrt{x^2 + b \cdot x}}{\sqrt{x^2}}\right)\)
Would you say that the following statement is correct?
"When multiplying a 'number' with infinite magnitude by zero, the result is unknown."
In other words, we say that the Zero-Product Property of Real Numbers does not apply when infinity is involved.
I understand that numbers with infinite magnitude cannot be "nailed down" as any specific real number, but (until now) I was thinking that a factor of zero would result in a product of zero, nonetheless.
(The concept of infinity was not discussed in any detail during my undergraduate years; nor do I see any such discussion in current precalculus texts. Perhaps precalculus courses should include a section devoted specifically to infinity, as opposed to having students pick up bits and pieces of information as they move on through calculus.)
Yes, I realized how simple it is after spending a few more minutes with the exercise during my coffee break this morning.
I feel somewhat sheepish that I did not think of factoring out an x from (x + a).
I'm not sure why, but I seem to be intimidated by radicals! Silly.
My first attempts at getting the (1 + a/x) form also failed because I have tunnel vision.
After I multiplied and divided the original expression by its conjugate, I went to multiply by sqrt(x^2)/sqrt(x^2). If I had written x/sqrt(x^2) instead, then I believe that I would have gone straight to the form posted by PKA.
Why did I not do this? Because my tunnel vision told me that a fraction is equal to one when the numerator LOOKS the same as the denominator, as opposed to HAVING the same value.
(Good grief.)
On the positive side, the guidance that I received from the contributors to this discussion has expanded my vision.
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