Can Conjugates Simplify Complex Limits as x Approaches Negative Infinity?

In summary: This is a reliable way to find limits if all of the terms in the limit have a finite limit and the final result doesn't have an indeterminant form like 0/0.
  • #1
cabron299
5
0
lim [3 + sqrt(x^2−x+1)]/[sqrt(9x^2−7x) −2x+5]
x−>-infinity

i know the conjugate of the nominator is 3-sqrt(x^2-x+1), but what is the conjugate of the denominator (or do i need to try some other method)?

PS- I cannot solve this with deravatives b/c the teacher tols us not to
 
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  • #2
I would handle the roots by writing, for example, sqrt(9x^2-7x)=3*|x|*sqrt(1-7x/(9x^2))=3*|x|*sqrt(1-7/(9x)). So the sqrt is 3*|x| times something whose limit is 1. DON'T mess with conjugates. DO keep track of the absolute values.
 
  • #3
ummm... my teacher said conjugates were the best for limits with a sqrt involved, i don't think we are in the same page (i wasnt able to understand what you did... if u could clarify that would be great =))
thanks in advance
 
  • #4
Conjugates aren't your best bet here. Too complicated. I'm basically just suggesting you divide numerator and denominator by x and look at the limit of each term. Be careful when you get to limits like sqrt(x^2-x+1)/x. What is the limit of that one?
 
  • #5
1?( sqrt(x^2-x+1)/(x) simplifying [x*sqrt(1-1/x+1/x^2)]/x, the x´s cancel and I am left off with sqrt(1-1/x+1/x^2)= 1 right?
 
  • #6
cabron299 said:
1?( sqrt(x^2-x+1)/(x) simplifying [x*sqrt(1-1/x+1/x^2)]/x, the x´s cancel and I am left off with sqrt(1-1/x+1/x^2)= 1 right?

Try it on your calculator. x->negative infinity. The limit is -1, right? The x comes out of the sqrt as |x|. |x|/x=(-1) if x is negative. I told you to be careful of that.
 
  • #7
oh ok i need to remember the absolute value after x comes out of the sqrt.. so after i have the limits of all the four parts i can sum them up?
thanks alot!
 
  • #8
cabron299 said:
oh ok i need to remember the absolute value after x comes out of the sqrt.. so after i have the limits of all the four parts i can sum them up?
thanks alot!

Yes. After you divide by x you don't get a 0/0 form for the limit. If you did you would need to think about using conjugates to cancel something. But here you don't.
 
  • #9
thanks a lot! i got the right answer (1/5) is this method trustworthy for most limits?
 
  • #10
If all the terms in your limit have a finite limit and the final result doesn't have an indeterminant form like 0/0, what could go wrong?
 

FAQ: Can Conjugates Simplify Complex Limits as x Approaches Negative Infinity?

How do I identify the type of limit?

To identify the type of limit, you need to look at the algebraic expression inside the limit. If it contains a variable in the denominator, it is a rational limit. If the variable is inside a square root, it is a radical limit. If the variable is in the exponent, it is an exponential limit. If the variable is in the trigonometric function, it is a trigonometric limit.

What are the steps to solve a rational limit?

To solve a rational limit, you need to factor the numerator and denominator, cancel out any common factors, and then plug in the limit value to evaluate the expression. If the limit value causes a division by zero, you need to apply the limit laws to rewrite the expression in a different form before plugging in the value again.

How do I simplify a radical limit?

To simplify a radical limit, you need to rationalize the numerator and denominator by multiplying by the conjugate of the radical expression. This will eliminate the radical and allow you to evaluate the limit using algebraic techniques.

What is the Squeeze Theorem and how is it used to solve limits?

The Squeeze Theorem states that if two functions, f(x) and g(x), are both approaching the same limit as x approaches a certain value, and f(x) is always less than or equal to g(x) for values of x close to the limit, then the limit of f(x) and g(x) will be the same. This theorem is useful when you have a limit expression that is not in a form that can be solved directly, but you can find two other functions that are approaching the same limit and use them to "squeeze" the desired limit.

Are there any special techniques for solving trigonometric limits?

Yes, there are several special techniques for solving trigonometric limits, such as using trigonometric identities, using the Squeeze Theorem, or converting the limit to an exponential form using Euler's formula. It is important to be familiar with these techniques and know when to apply them in order to solve trigonometric limits efficiently.

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