# Need Help with Homework: Question 1, 4 and 5

• Anakin_k
In summary, the conversation is about a person asking for help with a homework assignment consisting of 5 questions. They have completed all but the 4th question and would like someone to confirm their solutions for questions 1 and 5. Another person provides a detailed proof for question 4 and offers to help with questions 1 and 5, while reminding the original poster to follow the rules of not posting complete solutions.
Anakin_k
Hello everyone, I am in need of a little assistance. I have a homework assignment due soon that consists of 5 questions. Of which, I have done all but the 4th one. I did start on it but I'm not sure where to go from there. I also would like for someone to confirm my solutions for question 1 and 5, as I'm not too sure about them (I am confident with Q2 and Q3).

Here are the questions: http://i56.tinypic.com/30nb1xf.png

Question 1 solution: http://i52.tinypic.com/2hfht83.jpg
Question 5 solution: http://i56.tinypic.com/2jdhlzl.jpg
Question 4 beginning: http://i53.tinypic.com/15dvs6w.jpg

Thank you.

I can attempt to help on Question 4:

Proof:
Let $\epsilon > 0$. Since $\lim_{x \rightarrow \infty} g(x) = l$, there exists an $N_0$ such that if $x > N_0$, then $|g(x) - l| < \epsilon/2$ (*).

Similary, there exists an $N_1$ such that if $x > N_1$, then $|f(x)-m|<\epsilon/(2|-a|)$. It follows from the last inequality that $|-af(x)+am|<\epsilon/2$ (**).

Now choose $N=\max\{N_0, \, N_1\}$. So if $x > N$, inequalities (*) and (**) must hold. And adding (*) with (**), we get

$|g(x)-l| + |-af(x)+am| < \epsilon/2 + \epsilon/2 = \epsilon$.

But by the triangle inequality,

$|(g(x)-l) +(-af(x)+am)| \leq |g(x)-l| + |-af(x)+am|$.

Rearranging terms on the left-hand side of the above inequality, I think you get what you want: If $x > N$, then $|(g(x)-af(x))-(l-am)| < \epsilon$.

I hope this helps.

P.S. I forgot, this proof assumes $a$ does not equal zero. In the case it does equal zero, the problem reduces down to showing $\lim_{x \rightarrow}(g(x)-0f(x)) = \lim_{x \rightarrow}(g(x)-0) = \lim_{x \rightarrow}g(x) = l - 0m = l$, and this is already true since it's given.

Last edited:
That was extremely thorough Mathstatnoob, thank you for taking the time to post the answer. I appreciate it very much. Through your steps, I've learned how to manipulate little pieces here and there to arrive at the needed answer. :)

If anyone could confirm my Q1 and Q5 solutions, I'd be thankful.

Question 1 looks okay to me. for Q5, this is just the product lemma from the algebra of limits.

mathstatnoob said:
I can attempt to help on Question 4:

Proof:
Let $\epsilon > 0$. Since $\lim_{x \rightarrow \infty} g(x) = l$, there exists an $N_0$ such that if $x > N_0$, then $|g(x) - l| < \epsilon/2$ (*).

Similary, there exists an $N_1$ such that if $x > N_1$, then $|f(x)-m|<\epsilon/(2|-a|)$. It follows from the last inequality that $|-af(x)+am|<\epsilon/2$ (**).

Now choose $N=\max\{N_0, \, N_1\}$. So if $x > N$, inequalities (*) and (**) must hold. And adding (*) with (**), we get

$|g(x)-l| + |-af(x)+am| < \epsilon/2 + \epsilon/2 = \epsilon$.

But by the triangle inequality,

$|(g(x)-l) +(-af(x)+am)| \leq |g(x)-l| + |-af(x)+am|$.

Rearranging terms on the left-hand side of the above inequality, I think you get what you want: If $x > N$, then $|(g(x)-af(x))-(l-am)| < \epsilon$.

I hope this helps.

P.S. I forgot, this proof assumes $a$ does not equal zero. In the case it does equal zero, the problem reduces down to showing $\lim_{x \rightarrow}(g(x)-0f(x)) = \lim_{x \rightarrow}(g(x)-0) = \lim_{x \rightarrow}g(x) = l - 0m = l$, and this is already true since it's given.

Please re-read the Rules link at the top of the page. Posting solutions to homework questions is not permitted here. Instead, please provide hints, ask questions, find mistakes, etc.

## 1. What is the best way to approach Question 1?

The best way to approach Question 1 is to carefully read the question and identify the key concepts and information that it is asking for. Then, use your knowledge and understanding of the topic to formulate an answer. If you are still unsure, you can refer to your notes, textbook, or do some additional research to help you answer the question.

## 2. How do I solve Question 4?

Solving Question 4 will depend on the specific question and the subject it pertains to. Some tips for solving any homework question include breaking it down into smaller parts, using relevant formulas or equations, and double-checking your work for accuracy. If you are still struggling, you can reach out to your teacher or classmates for assistance.

## 3. Can I use outside resources for Question 5?

It is always best to check with your teacher or professor before using outside resources for your homework. They may have specific guidelines or restrictions on what sources are acceptable. If you are allowed to use outside resources, make sure to properly cite them and use them as a supplement to your own understanding and work.

## 4. What should I do if I don't understand a question?

If you don't understand a question, it is important to not panic. Take a deep breath and try to break down the question into smaller parts. You can also refer to your notes or textbook for clarification. If you are still struggling, don't be afraid to ask your teacher or classmates for help. It is better to seek assistance than to turn in an incorrect or incomplete answer.

## 5. How do I manage my time while completing these homework questions?

Effective time management is important when completing homework. Prioritize your assignments based on due dates and level of difficulty. Break down larger tasks into smaller chunks and set aside dedicated time to work on them. Also, make sure to take breaks and avoid distractions while working. If you are struggling with time management, consider creating a schedule or seeking advice from a teacher or academic advisor.

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