Limit as x approaches 0 for given function

  • #1

Homework Statement



lim x --> 0 for the function sqrt((1/x^2)-(1/x)) - sqrt((1/x^2)-(1/x))

Analyze the cases x > 0 and x < 0

The Attempt at a Solution



The solutions book simplifies the expression to

2x/(abs(x)*(sqrt(1+x)+sqrt(1-x)))

I know how to evaluate the limit from here. But I'm wondering why did they only place the absolute value sign for the single x in the denominator that multiplies with this (sqrt(1+x)+sqrt(1-x)) expression. Why isn't the absolute value sign around every single x?


The answer is that the limit doesnt exist.
 
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Answers and Replies

  • #2
575
76

Homework Statement



lim x --> 0 for the function sqrt((1/x^2)-(1/x)) - sqrt((1/x^2)-(1/x))

Analyze the cases x > 0 and x < 0

The Attempt at a Solution



The solutions book simplifies the expression to

2x/(abs(x)*(sqrt(1+x)+sqrt(1-x)))

I know how to evaluate the limit from here. But I'm wondering why did they only place the absolute value sign for the single x in the denominator that multiplies with this (sqrt(1+x)+sqrt(1-x)) expression. Why isn't the absolute value sign around every single x?


The answer is that the limit doesnt exist.

It looks like maybe your problem statement has a typo, but that doesn't seem to matter to the question that you asked.

The absolute value comes into play because ##\sqrt{x^2}=|x|##.
 
  • #3
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Why isn't the absolute value sign around every single x?
Why do you expect one?
 
  • #4
Ohh right I forgot out that equality. So you're saying all they did was sub in abs(x) wherever there as x^2?
 
  • #5
Ray Vickson
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Homework Statement



lim x --> 0 for the function sqrt((1/x^2)-(1/x)) - sqrt((1/x^2)-(1/x))

Analyze the cases x > 0 and x < 0

The Attempt at a Solution



The solutions book simplifies the expression to

2x/(abs(x)*(sqrt(1+x)+sqrt(1-x)))

I know how to evaluate the limit from here. But I'm wondering why did they only place the absolute value sign for the single x in the denominator that multiplies with this (sqrt(1+x)+sqrt(1-x)) expression. Why isn't the absolute value sign around every single x?


The answer is that the limit doesnt exist.

Your question must have a "typo" in it; what you have written is ##\sqrt{f(x)} - \sqrt{f(x)}##, which is identically 0 for all positive values of ##f(x) = 1/x^2 - 1/x.##
 
  • #6
35,514
11,979
Ohh right I forgot out that equality. So you're saying all they did was sub in abs(x) wherever there as x^2?
It just appears once. They replaced ##\sqrt{x^2}## by abs(x) (note the square root), because ##\sqrt{x^2} = |x|##.
 
  • #7
575
76
Ohh right I forgot out that equality. So you're saying all they did was sub in abs(x) wherever there as x^2?

No. I'm saying that they used the equality, ##\sqrt{x^2}=|x|##, as part of rewriting the given algebraic expression in way that is more easily analyzed in the context of this problem.

One could say that they replaced certain instances of ##\sqrt{x^2}## with instances of ##|x|##.

I don't like the word "substitute" to describe what is going on here. I can't exactly put my finger on why I don't like it. It just seems like the wrong word to use.
 

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