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: \(\displaystyle f(x)=frac{1}{x}\) there is one vertical asymptote at $x=\text{some value}$. Care to guess what that value is?
A horizontal asymptote is a line that a function approaches but never touches as the input values get larger or smaller. It represents the end behavior of a function.
To find the horizontal asymptote of a function, you can use the limit of the function as x approaches positive or negative infinity. If the limit is a finite number, that is the equation of the horizontal asymptote. If the limit is infinity or negative infinity, then the function does not have a horizontal asymptote.
Yes, it is possible for a function to have more than one horizontal asymptote. This can occur if the limit of the function as x approaches positive or negative infinity is a different finite number for each limit. In this case, the function will approach different lines as the input values get larger or smaller.
The horizontal asymptote of a function provides information about the end behavior of the function. It can help determine the overall shape of the function and whether it is increasing or decreasing in certain intervals. It is also useful in finding limits and evaluating the behavior of the function at extreme values.
No, a function cannot cross its horizontal asymptote. This is because the horizontal asymptote represents the end behavior of the function, and by definition, a function cannot cross its end behavior. If a function appears to cross its horizontal asymptote on a graph, it is likely due to a discontinuity or error in the graphing process.