HELP: Limit of absolute value

In summary, the limit of an absolute value function is the value that the function approaches as the input approaches a particular value. To find the limit, the function needs to be evaluated from both sides of the input value. A one-sided limit only considers values from one side while a two-sided limit considers values from both sides. The limit can be negative if the values approaching the input from both sides are negative and the function itself is negative. The limit is important for determining function behavior and solving equations involving absolute values.
  • #1
buffgilville
91
0
Compute the limit of: (absolute value)

f(x) = 4 (absolute value of x + 4) / (x+4)

as x approaches -4 from the right.

I got 4 because x>-4. Am I right?
 
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  • #2
Yep that's correct! When [tex]x > 4, |x + 4| = x + 4[/tex]
Thus,

[tex]lim_{x \rightarrow -4^+} f(x) = lim_{x \rightarrow -4^+} 4\frac{x+4}{x+4} = 4[/tex]
 
  • #3


Your answer is partially correct. The limit of the given function as x approaches -4 from the right is indeed 4. However, your reasoning is not entirely accurate. The reason why the limit is 4 is not because x is greater than -4, but rather because the absolute value of x+4 is always equal to 4 when x is approaching -4 from the right. This is because when x is approaching -4 from the right, it means that x is getting closer and closer to -4, but never actually reaches -4. And since the absolute value of a number is always positive, no matter how close x gets to -4, the absolute value of x+4 will always be 4. Therefore, the limit of the function is 4.
 

What is the limit of an absolute value function?

The limit of an absolute value function is the value that the function approaches as the input approaches a particular value.

How do you find the limit of an absolute value function?

To find the limit of an absolute value function, you need to evaluate the function from both sides of the input value and see if the values are approaching the same number. If they are, then that is the limit. If they are not, then the limit does not exist.

What is the difference between one-sided and two-sided limits for absolute value functions?

A one-sided limit for an absolute value function only considers the values approaching the input value from one side. A two-sided limit considers values approaching the input value from both sides, and the limit only exists if the values from both sides approach the same number.

Can the limit of an absolute value function be negative?

Yes, the limit of an absolute value function can be negative. This can happen when the values approaching the input value from both sides are negative and the absolute value function itself is negative.

Why is the limit of an absolute value function important?

The limit of an absolute value function is important because it can help determine the behavior of the function at a particular input value. It can also help with finding the derivatives of absolute value functions and solving certain types of equations involving absolute values.

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