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

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Homework Help Overview

The discussion revolves around finding the limit as x approaches 0 for the expression (1 - sqrt(x^2 - 1)) / x^2. Participants are exploring the implications of this limit in the context of calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • One participant attempts to manipulate the expression by multiplying by a conjugate, while another questions the algebraic steps taken. There is also a mention of graphical analysis leading to complex results, prompting a discussion about the validity of the limit.

Discussion Status

The discussion is ongoing, with participants providing feedback on each other's approaches. Some guidance has been offered regarding algebraic manipulation, and there is an exploration of different interpretations of the limit's behavior as x approaches 0.

Contextual Notes

Participants express uncertainty about the problem setup and whether the original problem statement was copied correctly. There is also a recognition that the limit may not exist based on the evaluations presented.

andrewjacobs
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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.
 
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(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?
 
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
 
andrewjacobs said:

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.
 

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