Find the Limit: x→0 of 1-sqrt(x^2-1)/x^2

[andrewjacobs]

Homework Statement

2. Find the limit.
lim
x->0
for
1-sqrt(x^2-1)
---------------
x^2

Homework Equations

No clue, that's why I am asking.

The Attempt at a Solution

I inverted the signs and multiplied by sqrt(x^1+1)+1
So:
sqrt(x^2+1)-1 sqrt(x^2+1)+1
--------------- * --------------
-(x^2) sqrt(x^2+1)+1
and came up with
x+1-1
----------------------
-(x^2)*sqrt(x^2+1)+1
which gave me
0/2

I am sure this isn't correct, but I do not know were I messed up.

 

[Dick]

(sqrt(x^2+1)-1)*(sqrt(x^2+1)+1)=(x^2+1)-1. Now isn't it? Not (x+1)-1. As (a+b)*(a-b)=a^2-b^2?

 

[Math Man900]

when i graphed it, i found the answer to be
infinity - infinity*i
or
(1-i)*infinity
none of those answer sound like what a teacher would be looking for, so i would guess the answer is either "limit does not exsist" or you copied the problem wrong

 

[HallsofIvy]

Homework Statement

2. Find the limit.
lim
x->0
for
1-sqrt(x^2-1)
---------------
x^2

Homework Equations

No clue, that's why I am asking.

The Attempt at a Solution

I inverted the signs and multiplied by sqrt(x^1+1)+1
So:
sqrt(x^2+1)-1 sqrt(x^2+1)+1
--------------- * --------------
-(x^2) sqrt(x^2+1)+1
and came up with
x+1-1
----------------------
-(x^2)*sqrt(x^2+1)+1
which gave me
0/2

I am sure this isn't correct, but I do not know were I messed up.

Your method is perfectly good. You algebra needs some work!
First you can't just change the x^2- 1 inside the square root into x^2+ 1 by "inverting the sign". Second, (1- sqrt(x^2+ 1))(1+ sqrt(x^2+ 1))= 1- (x^2+11)= -x^2.

In any case, if you just let x= 0 in the original form, you get (1- i)/0. Since the numerator is not 0, that limit does not exist.