Can you help me solve this small limits question?

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In summary, the conversation discusses the solution to a problem involving limits. The problem is to find the limit of (x^2*f(x) - 18)/(x - 3) given that the limit of (f(x) - 2)/(x - 3) is 7. One solution involves simplifying the expression and using the given limit to find the answer of 75. Another solution involves using the limit rules for derivatives to find the answer of 63. The conversation also addresses a misunderstanding of the first solution and clarifies the correct approach.
  • #1
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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  • #2
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
 
  • #3
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:
  • #4
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.
 
  • #5
<--- ashamed
 
  • #6
Thanks all,
But I think that I solved the question as the same as Soroban solved it (Look at my first post)

Thanks again.
 

1. What is a small limits question?

A small limits question is a type of research question that focuses on a specific aspect or detail within a larger topic or field of study. It often involves a narrow scope and can be answered through quantitative or qualitative methods.

2. How do you formulate a small limits question?

To formulate a small limits question, you need to identify a specific aspect or detail that you want to understand better within a larger topic. Then, you can use the "5 W's and 1 H" (who, what, where, when, why, and how) to guide your question and make it more precise.

3. What are the benefits of asking a small limits question?

Asking a small limits question allows for a more focused and in-depth analysis of a particular aspect, which can lead to a better understanding of the larger topic. It also helps to avoid broad and vague research questions, making the study more manageable.

4. What are some examples of small limits questions?

Examples of small limits questions include "What is the impact of social media on teenagers' self-esteem?" and "How does the use of technology affect employee productivity in the workplace?" These questions focus on a specific aspect within a larger topic and can be answered through research and analysis.

5. How does a small limits question differ from a broad research question?

A small limits question differs from a broad research question in terms of scope and specificity. A small limits question focuses on a narrow aspect within a larger topic, while a broad research question covers a broader range of topics. Additionally, a small limits question can be answered through data analysis, while a broad research question may require a more exploratory approach.

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