For lim as x -> -∞, how come lxl becomes -x?

  • Thread starter LearninDaMath
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In summary, as x approaches negative infinity, the absolute value function lxl approaches the negative value of x because it removes the negative sign from a number. Its purpose in this scenario is to ensure a positive value for lxl, making the representation of the limit simpler. The absolute value function can be used for other types of limits involving real numbers, such as trigonometric and exponential functions. It does not change the behavior of a limit, but rather affects how it is represented. The behavior of lxl is different as x approaches negative and positive infinity, with the output approaching -x and x respectively.
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LearninDaMath
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For lim as x --> -∞, how come lxl becomes -x?

Homework Statement



When you take the limit of a rational function as x approaches (-∞), I am dividing the numerator and denominator by the x value with highest exponent. Supposing the highest x value is √(x^2), so I divide the numerator and denominator by lxl. Then how come, in cases of x--> -∞, I am supposed to replace lxl with -x?

Does this question make sense?
 
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  • #2


Nevermind, it clicked literally 15 seconds after posting the question. No help needed on this particular question anymore :)
 

1. Why does lxl become -x as x approaches negative infinity?

As x approaches negative infinity, the absolute value function lxl will approach the negative value of x. This is because the absolute value function essentially removes the negative sign from a number, so as x becomes increasingly negative, the negative sign is removed and the value of lxl becomes closer to -x.

2. What is the purpose of the absolute value function in this scenario?

The absolute value function is used to ensure that the value of lxl remains positive, even when the input x is negative. This allows for a simpler and more concise representation of the limit as x approaches negative infinity.

3. Can the absolute value function be used for other types of limits?

Yes, the absolute value function can be used for other types of limits, as long as the input values are real numbers. It is commonly used in limits involving trigonometric functions and exponential functions.

4. How does the absolute value function affect the behavior of a limit?

The absolute value function does not change the behavior of a limit itself, but rather it ensures that the output of the limit remains positive. In other words, it does not affect the value of the limit, but rather the way it is represented.

5. Is the behavior of lxl as x approaches negative infinity different from its behavior as x approaches positive infinity?

Yes, the behavior of lxl is different as x approaches negative infinity compared to its behavior as x approaches positive infinity. As x approaches positive infinity, the absolute value function will result in the output approaching x, rather than -x as it does when x approaches negative infinity.

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