find the limit, as x goes to 1, of (1/x - 1) / (sqrt[x] - 1)

[jraed8]

\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, 1}\,\)\(\displaystyle \dfrac{\dfrac{1}{x}\, -\, 1}{\sqrt{\strut x\,}\, -\, 1}\)

the answer is 2 but i cannot get it:

. . . . .\(\displaystyle \dfrac{\dfrac{1}{x}\, -\, 1}{\sqrt{\strut x\,}\, -\, 1}\, \Rightarrow \, \dfrac{\dfrac{1}{x}\, -\, \dfrac{x}{x}}{\sqrt{\strut x\,}\, -\, 1}\, \Rightarrow\, \left(\,\dfrac{\left(\dfrac{1\, -\, x}{x}\right)}{\sqrt{\strut x\,}\, -\, 1}\,\right)\,\left(\dfrac{x}{x}\right)\, \Rightarrow\, \dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, 1}\)

. . . . .\(\displaystyle \Rightarrow\, \dfrac{1\, -\, x}{x\, \left(\sqrt{\strut x\,}\, -\, 1\right)}\, \Rightarrow\,\dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, x}\, \left(\, \dfrac{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}\, \right)\, \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x^3\, +\, x^2\, \sqrt{\strut x\,}\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}\)

. . . . .\(\displaystyle \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x^3\, -\, x^2}\, \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x}\)

. . . . .\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, 1}\,\)\(\displaystyle \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x}\, \Rightarrow\, \dfrac{1\, +\, 1\, -\, 1\, -\, 1}{1}\, =\, 0\)

I have tried multiplying conjugates/ lcd/ variations of both.

 

[Deleted member 4993]

\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, 1}\,\)\(\displaystyle \dfrac{\dfrac{1}{x}\, -\, 1}{\sqrt{\strut x\,}\, -\, 1}\)

the answer is 2 but i cannot get it:

. . . . .\(\displaystyle \dfrac{\dfrac{1}{x}\, -\, 1}{\sqrt{\strut x\,}\, -\, 1}\, \Rightarrow \, \dfrac{\dfrac{1}{x}\, -\, \dfrac{x}{x}}{\sqrt{\strut x\,}\, -\, 1}\, \Rightarrow\, \left(\,\dfrac{\left(\dfrac{1\, -\, x}{x}\right)}{\sqrt{\strut x\,}\, -\, 1}\,\right)\,\left(\dfrac{x}{x}\right)\, \Rightarrow\, \dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, 1}\)

. . . . .\(\displaystyle \Rightarrow\, \dfrac{1\, -\, x}{x\, \left(\sqrt{\strut x\,}\, -\, 1\right)}\, \Rightarrow\,\dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, x}\, \left(\, \dfrac{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}\, \right)\, \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x^3\, +\, x^2\, \sqrt{\strut x\,}\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}\)

. . . . .\(\displaystyle \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x^3\, -\, x^2}\, \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x}\)

. . . . .\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, 1}\,\)\(\displaystyle \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x}\, \Rightarrow\, \dfrac{1\, +\, 1\, -\, 1\, -\, 1}{1}\, =\, 0\)

I have tried multiplying conjugates/ lcd/ variations of both.

lim x->1 [(1/x)-1]/ [x^(1/2)-1]

= lim x->1 -[x-1]/ [x{x^(1/2)-1}]

= lim x->1 -[x^(1/2)+1]/ [x]

Continue....

 

[stapel]

\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, 1}\,\)\(\displaystyle \dfrac{\dfrac{1}{x}\, -\, 1}{\sqrt{\strut x\,}\, -\, 1}\)

the answer is 2 but i cannot get it:

. . . . .\(\displaystyle \dfrac{\dfrac{1}{x}\, -\, 1}{\sqrt{\strut x\,}\, -\, 1}\, \Rightarrow \, \dfrac{\dfrac{1}{x}\, -\, \dfrac{x}{x}}{\sqrt{\strut x\,}\, -\, 1}\, \Rightarrow\, \left(\,\dfrac{\left(\dfrac{1\, -\, x}{x}\right)}{\sqrt{\strut x\,}\, -\, 1}\,\right)\,\left(\dfrac{x}{x}\right)\, \)\(\displaystyle \color{red}{\Rightarrow\, \dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, 1}}\)

How did you multiply out x(sqrt[x] - 1) = x*sqrt[x] + x(-1) and get x*sqrt[x] - 1?

. . . . .\(\displaystyle \color{red}{\Rightarrow\, \dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, 1}}\)\(\displaystyle \Rightarrow\,\)\(\displaystyle \color{blue}{\dfrac{1\, -\, x}{x\, \left(\sqrt{\strut x\,}\, -\, 1\right)}}\,\)

In the denominator, how did you factor an x out of the 1?

Note: This and the previous question relate to errors which "cancelled out", leaving you with the correct result. This happenstance should not be assumed to work in generality! :shock:

. . . . .\(\displaystyle \Rightarrow\,\dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, x}\, \left(\, \dfrac{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}\, \right)\, \Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x^3\, +\, x^2\, \sqrt{\strut x\,}\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}\)

. . . . .\(\displaystyle \color{red}{\Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x^3\, -\, x^2}\, }\)\(\displaystyle \color{blue}{\Rightarrow\, \dfrac{x\, \sqrt{\strut x\,}\, +\, x\, -\, x^2\, \sqrt{\strut x\,}\, -\, x^2}{x}}\)

In the denominators, how did you get x³ - x² = x²(x - 1) equalling just x?

Note: With an "x - 1" remaining in the denominator, it is not yet useful to try to take the limit. However, by returning to an earlier stage of your working, we can do this:

. . . . .\(\displaystyle \dfrac{1\, -\, x}{x\,\sqrt{\strut x\,}\, -\, x}\, \left(\, \dfrac{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}{\left(x\, \sqrt{\strut x\,}\, +\, x\right)}\, \right)\, =\, \dfrac{(1\, -\, x)\, (x)\, \left(\sqrt{\strut x\,}\, +\, 1\right)}{x^3\, -\, x^2}\, \)

. . . . .\(\displaystyle =\, \dfrac{(1\, -\, x)\, (x)\, \left(\sqrt{\strut x\,}\, +\, 1\right)}{x^2\, (x\, -\, 1)}\, =\, \dfrac{(1\, -\, x)\, (x)\, \left(\sqrt{\strut x\,}\, +\, 1\right)}{-x^2\, (1\, -\, x)}\, =\, \dfrac{\sqrt{\strut x\,}\, +\, 1}{-x}\)

Now can you see where to go?