Understanding Limits: Does it Exist or Go to Infinity?

In summary, the concept of a limit is a fundamental concept in mathematics that deals with the behavior of a function as its input values get closer and closer to a specific value. The existence of a limit can be determined by evaluating the function at values approaching the specific value from both the left and right sides. A finite limit is when the function approaches a specific value as its input values get closer to a certain value, while a limit at infinity is when the function approaches a specific value as its input values get larger and larger in magnitude. A function can have a limit at a point where it is not defined, but the limit may not exist if the function has a discontinuity at that point. Limits are used in real-world applications, such as
  • #1
Punkyc7
420
0
How can you tell if a limit does not exit or that it goes to infinity?

examplelim[itex]\underbrace{x\rightarrow}_{0}[/itex]([itex]\frac{\sqrt{x+1}}{x}[/itex])

The x goes to 0The book says the limit is [itex]\infty[/itex] but if you take the left side limit you get -[itex]\infty[/itex] and if you take the right side of the limit you [itex]\infty[/itex]. So wouldn't this limit not exist, because you have two different limits?
 
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  • #2
Yes, I agree with you. I think there's a typo somewhere.
 
  • #3
That's what I thought, thank you for looking at it.
 

FAQ: Understanding Limits: Does it Exist or Go to Infinity?

What is the concept of a limit?

The concept of a limit is a fundamental concept in mathematics that deals with the behavior of a function as its input values get closer and closer to a specific value. It helps us understand what happens to a function when its input values approach a certain value, even if the function is not defined at that value.

How do you determine the existence of a limit?

The existence of a limit can be determined by evaluating the function at values approaching the specific value from both the left and right sides. If the values approach the same number, then the limit exists. However, if the values approach different numbers or if the function has a discontinuity at that value, then the limit does not exist.

What is the difference between a finite limit and a limit at infinity?

A finite limit is when the function approaches a specific value as its input values get closer and closer to a certain value. A limit at infinity, on the other hand, is when the function approaches a specific value as its input values get larger and larger in magnitude, either positive or negative. In other words, the function has no bound on its input values.

Can a function have a limit at a point where it is not defined?

Yes, a function can have a limit at a point where it is not defined. This is because the concept of a limit allows us to understand the behavior of a function at a specific value, even if the function is not defined at that value. However, the limit may not exist if the function has a discontinuity at that point.

How are limits used in real-world applications?

Limits are used in various real-world applications, such as in physics, engineering, and economics. For example, limits are used to understand the behavior of a moving object as it approaches a certain point, or to analyze the growth of a population as it approaches a maximum value. In economics, limits are used to calculate marginal costs and marginal revenues, which are important in decision-making processes.

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