Find Limit of x Approaching 4 - Problem Solved

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In summary, the conversation is about finding the limit of a function as x approaches 4, where the function is 3x + square root of x-4. The limit is determined by taking the limit on both sides, and if they are equal, that is the limit. However, if the function does not exist on one side, then the one-sided limit is the limit. It is important to clarify with the professor which interpretation of limit should be used. Additionally, it is important to use the correct terminology, as "limit superior" has a different meaning.
  • #1
Strelka
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I have a question about the limit problem. I need find limit if it exists.

lim x (approach to 4) (3x+square root x-4).
If I plug number for x=4, I get 12. But I am concern about square root. Do I need find one-sided limit from x approach to 4+ and x approach to 4-. Or 12 will be the correct answer?
Thanks, everyone
 
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  • #2
If it's not specified,then the superior limit is understood.

Daniel.
 
  • #3
Usually the limit (in the case of real variables) is defined by getting limits on both sides. If they are equal, that is the limit. If not, the limit does not exist.
 
  • #4
Except when it is clear that the function itself does not exist on one side. Then the one sided limit is the limit.
 
  • #5
We both assumed the limit & the function were defined on the reals (real intervals).

Daniel.
 
  • #6
Limit Laminant...


[tex]\boxed{\lim_{x \rightarrow 4} 3x + \sqrt{x - 4} = 12}[/tex]

 
  • #7
You should ask your professor how he would like you to answer. However, I think you should take the strict interpretation -- it's written as a two-sided limit, so don't assume it's a one-sided limit.

When writing your answer, you can always say the limit doesn't exist, and then write down what the one-sided limit is.


P.S. dex: I know everyone knows what you meant, but "limit superior" (written "lim sup") means something else.
 
  • #8
Yes, I would use the definition. By the definition, if you're working over the reals, the limit doesn't exist. As Hurkyl said, putting down the one-sided limit as well couldn't hurt. But unless it has been clearly stated in class that you have a different definition for limits when the function doesn't exist on one side, then you shouldn't assume so.
 
  • #9
Thanks guys a lot
 

FAQ: Find Limit of x Approaching 4 - Problem Solved

What is the concept of finding the limit of x approaching 4?

The concept of finding the limit of x approaching 4 is a fundamental concept in calculus. It involves determining the value that a function approaches as the independent variable (x) gets closer and closer to a specific value (4 in this case). This value is known as the limit and is used to describe the behavior of a function near a given point.

How is the limit of x approaching 4 solved?

To solve the limit of x approaching 4, we use the rules and properties of limits, such as the sum, difference, product, and quotient rules. We also use algebraic manipulation and substitution to simplify the expression and evaluate the limit. If the limit is indeterminate (cannot be evaluated directly), we use techniques such as L'Hôpital's rule to find the limit.

Can the limit of x approaching 4 be different on each side?

Yes, the limit of x approaching 4 can be different on each side. This means that the function may approach different values as x approaches 4 from the left and from the right. In such cases, the limit does not exist, and we say that the function has a discontinuity at x = 4.

What is the significance of finding the limit of x approaching 4?

Finding the limit of x approaching 4 helps us understand the behavior of a function near a given point. It can also help us determine if a function is continuous at that point. Additionally, limits are essential in calculus as they are used to define derivatives and integrals, which are crucial concepts in many areas of science and engineering.

Can the limit of x approaching 4 be calculated for all functions?

Not necessarily. The limit of x approaching 4 can only be calculated if the function is defined and continuous at x = 4. If the function has a discontinuity at x = 4, the limit does not exist. Additionally, some functions may have limits that cannot be evaluated using algebraic methods, in which case, we use numerical or graphical methods to approximate the limit.

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