Solving Limit for Horizontal Asymtotes

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Discussion Overview

The discussion revolves around understanding horizontal and vertical asymptotes in the context of limits, particularly as they relate to the function $$\frac{1}{x}$$. Participants explore the definitions, conditions, and examples of horizontal asymptotes, as well as the criteria for identifying vertical asymptotes.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant states that the limit $$\lim_{{x}\to{\infty}} \frac{1}{x}$$ approaches y = 0, questioning how this indicates a horizontal asymptote.
  • Another participant explains that horizontal asymptotes are determined by limits as $$x \rightarrow \infty$$ or $$x \rightarrow -\infty$$, noting that they represent values a graph approaches but never reaches.
  • It is mentioned that there are rules for calculating horizontal asymptotes based on the degrees of the numerator and denominator.
  • A participant provides an example of another limit, $$\lim_{{x}\to{\infty}} \frac{x}{x^2+1}$$, which also approaches 0, illustrating the concept further.
  • Questions arise about identifying vertical asymptotes, with one participant suggesting they occur when the denominator equals zero, using $$f(x)=\frac{1}{x}$$ as an example.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and conditions for horizontal and vertical asymptotes, but the discussion includes questions and clarifications rather than a consensus on specific examples or rigorous proofs.

Contextual Notes

Some participants express confusion regarding their lecture notes, indicating potential gaps in understanding or clarity about the concepts discussed.

Who May Find This Useful

Students studying calculus, particularly those learning about limits and asymptotic behavior in functions.

tmt1
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We had to solve this limit

$$\lim_{{x}\to{\infty}} \frac{1}{x}$$

the answer is y= 0 is the Horizontal Asymptotes. I get the y = 0 but how do we know that it is the horizontal asymptote?
 
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I don't know how rigorous of an argument you are looking for but it's important to note that horizontal asymptotes only occur when we are looking at limits as $x \rightarrow \infty$ or $x \rightarrow -\infty$. As you know they are a value that a graph is forever leaning towards but never really reaches. We can't directly solve for this limit like we can for many other ones, but we see the value it is tending towards.

The limit you posted is a very standard starting point for looking at horizontal asymptotes. You will see that there are some rules for calculating them that have to do with the degree of the numerator compared to the degree of the denominator.

Just to show you another example, if you look at:

$\displaystyle \lim_{{x}\to{\infty}} \frac{x}{x^2+1}$

the answer is also 0 since the denominator grows faster than the numerator. Anyway, can you tell me what kind of insight you are looking for specifically for the problem you gave and maybe I can better explain it? :)
 
Jameson said:
I don't know how rigorous of an argument you are looking for but it's important to note that horizontal asymptotes only occur when we are looking at limits as $x \rightarrow \infty$ or $x \rightarrow -\infty$. As you know they are a value that a graph is forever leaning towards but never really reaches. We can't directly solve for this limit like we can for many other ones, but we see the value it is tending towards.

The limit you posted is a very standard starting point for looking at horizontal asymptotes. You will see that there are some rules for calculating them that have to do with the degree of the numerator compared to the degree of the denominator.

Just to show you another example, if you look at:

$\displaystyle \lim_{{x}\to{\infty}} \frac{x}{x^2+1}$

the answer is also 0 since the denominator grows faster than the numerator. Anyway, can you tell me what kind of insight you are looking for specifically for the problem you gave and maybe I can better explain it? :)

Thanks I think that is what I needed to know, I was just confused by some of my notes from the lecture. How can you tell if a limit has Vertical Asymptotes?
 
Vertical asymptotes occur when you divide by 0 essentially. So if we use the example of: $$f(x)=\frac{1}{x}$$ there is one vertical asymptote at $x=\text{some value}$. Care to guess what that value is?
 
Jameson said:
Vertical asymptotes occur when you divide by 0 essentially. So if we use the example of: $$f(x)=frac{1}{x}$$ there is one vertical asymptote at $x=\text{some value}$. Care to guess what that value is?
When x = 0. Thanks, so if there is a vertical asymptote for a function, it occurs when the denominator is equal to 0?
 

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