Is the Limit as x Approaches 2 of ((1/x)-(1/2))/(x-2) Zero?

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In summary, the limit of ((1/x)-(1/2))/(x-2) as x approaches 2 is equal to 0. However, it is incorrect to say that 0/0 equals 0 until it is proven. The expression can be simplified to (2-x)/((2x)*(x-2)), and the second factor is equal to -1. The derivative of x/(2x-1) is not equal to zero unless x=0.
  • #1
TayTayDatDude
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Homework Statement


lim
x[tex]\rightarrow[/tex]2 ((1/x)-(1/2))/(x-2)


Homework Equations





The Attempt at a Solution


Limit x-> for all of them, too tedious to keep rewriting it
=((2/2x)-(x-2x))/(x-2)
=((2-x)/(2x))/(x-2)
=((2-x)/(2x)) * (1/(x-2))
=(2-x)/(2x^2-4x)
sub 2 in = 0/0 = 0

therefore when x approaches 2, y approaches 0?

=
 
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  • #2


Never say 0/0=0. Never, until you have actually shown it. You have (2-x)/((2x)*(x-2)). (2-x)/(x-2)=-(x-2)/(x-2)=(-1). Now what's the limit?
 
  • #3


Dick said:
Never say 0/0=0. Never, until you have actually shown it. You have (2-x)/((2x)*(x-2)). (2-x)/(x-2)=-(x-2)/(x-2)=(-1). Now what's the limit?


How did you go from (2-x)/((2x)*(x-2)) to (2-x)/(x-2)?
 
  • #4


I didn't. I just separated (2-x)/((2x)*(x-2)) into [1/(2x)] * [(2-x)/(x-2)] and made the observation that the second factor is -1.
 
  • #5


Dick said:
I didn't. I just separated (2-x)/((2x)*(x-2)) into [1/(2x)] * [(2-x)/(x-2)] and made the observation that the second factor is -1.

So then it is 0?

Btw, does the derivative of x/(2x-1) = 0?
 
  • #6


TayTayDatDude said:
So then it is 0?

Btw, does the derivative of x/(2x-1) = 0?

Is what 0? No, the derivative of x/(2x-1) is not zero. Unless x=0.
 

1. What is a math limit problem?

A math limit problem is a type of mathematical problem that involves finding the value that a function approaches as the input approaches a certain value. This value is called the limit.

2. What is the purpose of solving a math limit problem?

The purpose of solving a math limit problem is to understand the behavior of a function as the input approaches a certain value. This can help in evaluating functions, determining continuity, and finding maximum and minimum values of a function.

3. How do I solve a math limit problem?

To solve a math limit problem, you can use various techniques such as direct substitution, factoring, rationalization, and L'Hopital's rule. It is important to also understand the properties of limits and use them to simplify the problem.

4. What are some common types of math limit problems?

Some common types of math limit problems include finding limits of rational functions, trigonometric functions, exponential and logarithmic functions, and piecewise functions.

5. Can math limit problems have more than one solution?

Yes, math limit problems can have multiple solutions. This can happen when the function has different behaviors on either side of the limit value or when the limit does not exist.

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