Precise definition for the limit

In summary, the conversation discusses the process of finding values of delta that correspond to a given epsilon, in the context of solving the equation (4x+1)/(3x-4)=4.5 as the limit approaches 2. The goal is to find an interval around 2 in which the fraction will always fall between 4 and 5, making the solution easier to obtain. The use of a calculator is also mentioned, but it is emphasized that there is more to the solution than just plugging in numbers.
  • #1
realism877
80
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(4x+1)/(3x-4)=4.5 lim approaches 2

Find values of delta that correspond to epsilon=0.5

I know how to use the calculator to get the solution. I just want to know how to get rid of that nasty fractions to make things a lot easier.
 
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  • #2
realism877 said:
(4x+1)/(3x-4)=4.5 lim approaches 2

Find values of delta that correspond to epsilon=0.5

I know how to use the calculator to get the solution.
I'm not so sure. There's more to this than just plugging numbers into a calculator.

What you need to do is to find an interval around 2, (2 - δ, 2 + δ), so that no matter what x you pick in the aforementioned interval, (4x + 1)/(3x - 4) is between 4.5 - .5 and 4.5 + .5. IOW, between 4 and 5.
realism877 said:
I just want to know how to get rid of that nasty fractions to make things a lot easier.
 

FAQ: Precise definition for the limit

What is the definition of a limit?

The definition of a limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It is denoted by the notation lim f(x), where x represents the input value and f(x) represents the function.

How is the limit of a function calculated?

The limit of a function is calculated by evaluating the function at values that are very close to the desired input value. As these values get closer and closer to the input value, the resulting outputs can be used to determine the limit of the function.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function as the input approaches the desired value from one direction (either from the left or the right). A two-sided limit takes into account the behavior of the function from both directions as the input approaches the desired value.

What does it mean if a function has no limit at a specific value?

If a function has no limit at a specific value, it means that the behavior of the function as the input approaches that value is either undefined or infinite. This could be due to a vertical asymptote, a discontinuity, or simply a value that is not in the domain of the function.

How can knowing the limit of a function be useful?

Knowing the limit of a function can be useful in many mathematical applications, such as finding the maximum or minimum value of a function, determining the continuity of a function, and evaluating limits in calculus problems. It can also help with understanding the behavior and properties of a function.

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