Solving Simple Limit Problem Without Substitution

  • Thread starter Thread starter es
  • Start date Start date
  • Tags Tags
    Limit
Click For Summary

Homework Help Overview

The discussion revolves around evaluating the limit of the expression (1-Sqrt(x-2))/(x-3) as x approaches 3, within the context of calculus. Participants explore various methods to approach the problem without relying solely on substitution.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster discusses a substitution method they used while tutoring, prompting questions about alternative approaches. Some participants suggest multiplying by the conjugate and mention L'Hospital's rule as a potential method. There is also a moment of reflection from one participant regarding a misunderstanding of the problem's components.

Discussion Status

The discussion is active, with multiple participants offering different strategies for solving the limit problem. While no consensus has been reached, various methods are being explored, and participants are engaging with each other's ideas.

Contextual Notes

There is an indication that the original poster's student has not yet covered certain concepts in class, which may influence their understanding and acceptance of the methods discussed.

es
Messages
70
Reaction score
0
I tutor high school students in Calc and the other day I came across this problem.

Limit of (1-Sqrt(x-2))/(x-3) as x->3

I tried coaching the student on how to simplify the expression and in the end I just showed him this substitution.

Let u=Sqrt(x-2)

Then

(1-Sqrt(x-2))/(x-3) = (1-u)/(u^2-1)

And the Limit becomes

Limit of (1-u)/(u^2-1)=-1/(u+1) as u->1 which is -1/2

He looked at me like I had just done some black magic. I explained substitution to him and why it worked, showed him a couple of other simple examples, and confirmed the answer numerically (like they do in basic calc books when the limit concept is first presented). I still don't think he is 100% convinced because they had not covered this in his class yet which leads to my question.

Can the original problem be solved without substitution?
 
Physics news on Phys.org
Well, you might do it this way:
\frac{(1-\sqrt{x-2})}{x-3}=\frac{(1-\sqrt{x-2})}{x-3}*1=\frac{(1-\sqrt{x-2})}{x-3}*\frac{(1+\sqrt{x-2})}{(1+\sqrt{x-2})}=\frac{3-x}{(1+\sqrt{x-2})*(x-3)}=-\frac{1}{(1+\sqrt{x-2})}
And so on..
 
Multiplying by the conjugate would do the trick, but personally, I feel the easiest way is L'Hospital's rule .
 
Yipe. For some reason I though 1-x would be the numerator. Boy do I feel sheepish. :) Thanks for the help!
 

Similar threads

  • · Replies 12 ·
Replies
12
Views
2K
  • · Replies 12 ·
Replies
12
Views
2K
  • · Replies 58 ·
2
Replies
58
Views
5K
  • · Replies 11 ·
Replies
11
Views
2K
  • · Replies 22 ·
Replies
22
Views
3K
  • · Replies 31 ·
2
Replies
31
Views
3K
  • · Replies 8 ·
Replies
8
Views
2K
Replies
16
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
Replies
4
Views
2K