What Is the Correct Limit Calculation for limx->3 (x^2f(x) - 18) / (x-3)?

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Discussion Overview

The discussion revolves around the limit calculation for the expression limx→3 (x2f(x) - 18) / (x - 3), given that limx→3 (f(x) - 2) / (x - 3) = 7. Participants explore different approaches to solving the limit and question the validity of their reasoning.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes that since the denominator approaches zero and the limit exists, the numerator must also approach zero, leading to the conclusion that f(x) = 2.
  • Another participant questions this reasoning, suggesting that the limit resembles a difference quotient for derivatives and proposes that limx→3 x2f'(3) = 63.
  • A different participant provides a method of rewriting the limit expression by adding and subtracting 2x2 to facilitate simplification, ultimately arriving at the same result of 75.
  • Some participants express uncertainty about the correctness of their approaches and calculations, particularly regarding the implications of the limit existing.
  • One participant acknowledges a potential error in their reasoning and seeks clarification on the correct interpretation of the limit.

Areas of Agreement / Disagreement

There is no consensus on the correct approach to the limit calculation. Multiple competing views and methods are presented, with participants questioning each other's reasoning without reaching a definitive conclusion.

Contextual Notes

Participants express uncertainty about the implications of the limit existing and the assumptions underlying their calculations. There are unresolved mathematical steps and differing interpretations of the limit's behavior.

Zargawee
[SOLVED] Small Limits Question

Hi There,
I have this Simple Question In Limits :

If limx->3 (f(x) - 2) / (x - 3) = 7
Then limx->3 (x2f(x) - 18) / (x-3) = ??

I solved the question this way :
Since the denominator equals Zero , and limit exists , then the numerator equals zero .
[4]f(x) - 2 = 0 ---> f(x) = 2

limx->3 (x2f(x) - 18) / (x-3) =
limx->3 (2x2 - 18) / (x-3) =
limx->3 2(x2- 9) / (x-3) =
limx->3 2(x-3) (x+3) / (x-3) =
limx->3 2(x+3) =
2 (3+3) = 12

But I also solved in this way :

limx->3 (x2f(x) - 18) / (x-3) =
limx->3 (x2f(x) - 2x2 + 2x2 - 18) / (x-3)
limx->3 (x2f(x) + (2x2) /(x-3) - 2x2 - 18) / (x-3)
limx->3 (x2) (f(x) + 2) /(x-3) - 2(x2 - 9) / (x-3)
limx->3 (x2) (f(x) + 2) /(x-3) - 2(x2 - 9) / (x-3)
limx->3 (x2) 7 - limx->3 2(x-3)(x+3) / (x-3)
limx->3 (x2) 7 - limx->3 2(x-3)(x+3) / (x-3)
32 * 7 + limx->3 (x+3)
(9 * 7) + (3 + 3) = 63 + 12 = 75


What's the wrong with the first One ?
Please help.
 
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Originally posted by Zargawee
Since the denominator equals Zero , and limit exists , then the numerator equals zero .
[4]f(x) - 2 = 0 ---> f(x) = 2

What's the wrong with the first One ?
Please help.
I'm not sure about this reasoning. Your limit looks very similar to a difference quotient for derivatives. You could write limx-->a(f(x)-f(a))/(x-a)=f'(a). In this case, f'(3)=7, f(3)=2. You want to find limx-->ax2*f'(a)=a2f'(a)=9*7=63
 
Hello, Zargawee!

I found an answer. (Please check my work!)

We are given: limx->3 [f(x) - 2]/[x - 3] = 7

We're asked to find: limx->3[x2f(x) - 18]/[x - 3]

In the numerator, add 2x2 and subtract 2x2:
x2f(x) - 18 + 2x2 - 2x2 = x2[f(x) - 2] + 2(x2 - 9)

We have: x2[f(x) - 2]/(x - 3) + 2(x2 - 9)/(x - 3)

The second term reduces: 2(x - 3)(x + 3)/(x - 3) = 2(x + 3)

Then we have: x2[f(x) - 2)/(x - 3) + 2(x + 3)

Taking limits, we have:
limx->3(x^2) * limx->3 [f(x) - 2]/(x - 3) + limx->3 2(x + 3)

We are given that the middle limit is 7.

Therefore, the answer is: (32)(7) + 2(6) = 75
 
Last edited by a moderator:
Hi There,
I have this Simple Question In Limits :

If limx->3 (f(x) - 2) / (x - 3) = 7
Then limx->3 (x2f(x) - 18) / (x-3) = ??

I solved the question this way :
Since the denominator equals Zero , and limit exists , then the numerator equals zero .
f(x) - 2 = 0 ---> f(x) = 2

That's not quite right; what is true is that

limx→3 f(x) = 2


Soroban's answer looks right.
 
<--- ashamed
 
Thanks all,
But I think that I solved the question as the same as Soroban solved it (Look at my first post)

Thanks again.
 

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