Question

Calculus: Evaluate lim as x->1 of (1- (1/x))/(1-(1/sqrt(x)))

Answer 1

\[\large \frac{1-\frac{1}{x}}{1-\frac{1}{\sqrt{x}}}\]?

Answer 2

Yes @satellite73

Answer 3

Is the answer is 2?

Answer 4

According to wolfram alpha, yes the answer is 2, but please help me how to approach this problem?

Answer 5

idk i didn't to it
the first obvious thing to do since it is of the form \(\frac{0}{0}\) is to use l'hopital, but that is probably no allowed and it is not necessary
the next thing would be to get rid of the compound fraction

Answer 6

or use wolfram lol

Answer 7

Nope l'hopital at this point is no good haha

Answer 8

can you think of anything to sandwich this with?

Answer 9

k then get rid of the compound fraction

Answer 10

no sandwich, it is all algebra

Answer 11

First,simplify the numerator and denominator.

Answer 12

okay.. I'll try getting rid of the compound fraction then

Answer 13

multiply top and bottom by \(x\)

Answer 14

then probably it would be a good idea to rationalize the denominator

Answer 15

Rationalise the denominator.

Answer 16

isnt the second step\[\frac{x-1}{x-\frac{x}{\sqrt(x)}}\] ?

Answer 17

good question!
yes but \[\frac{x}{\sqrt{x}}=\sqrt{x}\]

Answer 18

oh wow.. I learn new things everyday haha, thanks for that

Answer 19

you are four steps away
1) rationalize the denomiator
2) factor the denominator
3) cancel (which was the whole point of this)
4) plug in 1

Answer 20

ah you knew
\[\frac{x}{\sqrt{x}}=\sqrt{x}\] already
either subtract the exponents, or else square \(\frac{x}{\sqrt{x}}\) and see that you get \(x\)

Answer 21

did you complete step one yet? rationalize the denominator?

Answer 22

I've actually successfully evaluated the limit. Thanks so much!!

Answer 23

And no it wasn't obvious at first that x/sqrt(x) is just sqrt(x)

Answer 24

yw
hope it was clear
gimmick is to cancel \(x-1\) to get rid of the 0 top and bottom

Answer 25

I had to evaluate it to see for myself haha. thanks!