The limit $$\lim_{{x}\to{\infty}} \frac{1}{x}$$ results in a horizontal asymptote at y = 0, as confirmed by the discussion. Horizontal asymptotes are determined by evaluating limits as x approaches infinity or negative infinity, indicating values that graphs approach but never reach. The discussion also highlights that the degree of the numerator compared to the denominator is crucial in determining horizontal asymptotes, exemplified by $$\lim_{{x}\to{\infty}} \frac{x}{x^2+1}$$, which also approaches 0. Vertical asymptotes occur when the denominator equals zero, as illustrated by the function $$f(x)=\frac{1}{x}$$, which has a vertical asymptote at x = 0.
PREREQUISITESStudents studying calculus, mathematics educators, and anyone seeking to deepen their understanding of asymptotic behavior in functions.
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? :)
When x = 0. Thanks, so if there is a vertical asymptote for a function, it occurs when the denominator is equal to 0?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?