Flipping signs for limits at negative infinity

In summary, the conversation discusses a question regarding the limit of a function as x approaches negative infinity. The correct answer is -3 and there is a proof that involves substituting -u for x. However, there is also a mistake in the conversation.
  • #1
farleyknight
146
0

Homework Statement



[itex]\lim_{x \to -\infty} x + \sqrt{x^2 + 6x}[/itex]

Homework Equations


The Attempt at a Solution



Previous attempt was guessing it was [itex]\infty[/itex], but I see now my flaw and the actual answer is -3. Somewhere else on the web, might have been this forum, it was said that one could flip the sign and get

[itex]\lim_{x \to -\infty} x + \sqrt{x^2 + 6x} = \lim_{x \to \infty} -x + \sqrt{x^2 + 6x}[/itex]

Which I can see intuitively, since the what is under the radical would be positive either way, which implies that the sign only need be flipped for x. However, is there a generalized proof that includes any number of polynomials and roots, for this fact?

Thanks,
- Farley
 
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  • #2
What you are doing is substituting -u for x. Then as x->-infinity, u->+infinity. But x+sqrt(x^2+6x) turns into -u+sqrt(u^2-6u), doesn't it?
 
  • #3
Yeah, that's probably a mistake.. But that does explain it. Thanks.
 

1. What is the purpose of "flipping signs" for limits at negative infinity?

Flipping signs is a technique used to evaluate limits at negative infinity, which involves changing the sign of the variable being approached by negative infinity. This allows us to simplify the expression and find the limit more easily.

2. When do we need to use the "flipping signs" technique for evaluating limits at negative infinity?

The "flipping signs" technique is typically used when the expression being evaluated involves a fraction with a variable in the denominator. In this case, flipping the sign of the variable allows us to eliminate the fraction and evaluate the limit more easily.

3. How do we perform the "flipping signs" technique for evaluating limits at negative infinity?

To perform the "flipping signs" technique, we simply change the sign of the variable being approached by negative infinity. For example, if the limit is approaching -∞, we change any x values to -x before evaluating the expression.

4. Can we use the "flipping signs" technique for evaluating limits at positive infinity?

No, the "flipping signs" technique is only used for evaluating limits at negative infinity. For limits at positive infinity, we use different techniques such as dividing by the highest power of the variable or using L'Hopital's rule.

5. Are there any restrictions to using the "flipping signs" technique for evaluating limits at negative infinity?

Yes, the "flipping signs" technique only works for limits at negative infinity where the variable is approaching a constant value. If the variable is approaching a function, another technique must be used to evaluate the limit.

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