# How to rewrite limit to prove?

• ianq
In summary, the conversation is discussing a limit problem in calculus class where the limit is equal to -1. The professor suggests rewriting the limit in the form (x-3)g(x) and finding g(x). The person is unsure of how to find g(x) and the conversation suggests doing algebra to simplify the equation and potentially find a factor.
ianq
Hi,
I'm taking calculus I in college right now and for some reason we stated with limits...We're giving the following limit (sorry, I don't know how to work the board's code to make it look pretty):

lim (x->3) is (x+6) / (x^4 - 4x^3 + x^2 + x + 6) = -1

The prof suggested we rewrite lim (x->3) is (x+6) / (x^4 - 4x^3 + x^2 + x + 6) + 1 = 0 in the form (x-3)g(x) and find g(x). Any idea what g(x) would be and how to find it?

Did you ever think about just plugging in 3 for x and seeing what happens?

Yep, I did. I get -1 = -1. But considering it's a class exercise and the prof wants us to rewrite it as (x-3)g(x) I'm clueless...

ianq said:
Hi,
I'm taking calculus I in college right now and for some reason we stated with limits...We're giving the following limit (sorry, I don't know how to work the board's code to make it look pretty):

lim (x->3) is (x+6) / (x^4 - 4x^3 + x^2 + x + 6) = -1

The prof suggested we rewrite lim (x->3) is (x+6) / (x^4 - 4x^3 + x^2 + x + 6) + 1 = 0 in the form (x-3)g(x) and find g(x). Any idea what g(x) would be and how to find it?
so do the algebra! What is
$$\frac{x+6}{x^4- 4x^3+ x^2+ x+ 6}+ 1$$?

I assume you know that is the same as
$$\frac{x+6}{x^4- 4x^3+ x^2+ x+ 6}+ \frac{x^4- 4x^3+ x^2+ x+ 6}{x^4- 4x^3+ x^2+ x+ 6}$$

Add and try to factor the numerator. If you already know one factor, that should be easy!

## 1. How do I rewrite a limit expression?

To rewrite a limit expression, you can use algebraic manipulation or apply known limit laws. For example, you can multiply the numerator and denominator by a common factor or use the limit of a sum or difference rule.

## 2. What is the purpose of rewriting a limit expression?

Rewriting a limit expression can help simplify the expression and make it easier to evaluate. It can also help you apply limit laws and solve more complex limit problems.

## 3. Can I rewrite any limit expression?

Yes, you can rewrite any limit expression as long as it follows the basic rules and properties of limits. However, some limit expressions may require more advanced techniques or may not have a definitive solution.

## 4. How do I prove a limit using rewriting?

To prove a limit using rewriting, you need to manipulate the expression until it matches a known limit form. You can then use the limit laws and substitution to evaluate the limit and show that it equals the desired value.

## 5. Are there any common mistakes to avoid when rewriting a limit expression?

Yes, some common mistakes to avoid when rewriting a limit expression include forgetting to apply the limit laws, making algebraic errors, and assuming that a limit exists when it does not. It is important to carefully follow the steps and double-check your work to avoid these mistakes.

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