Recent questions in Asymptotes
imire37 2022-08-13

### Asymptotes for a function$f\left(x\right)=\frac{{x}^{2}-2ax}{x-a}$where a is positive.The question asks to give equations for the 2 asymptotes. The first is obvious: x=a. The other asymptote is y=x−a but I am unsure how to derive this.

Dillan Valenzuela 2022-08-11

### Question regarding the Asymptotes of HyperbolaTake a curve ,some curve which converges to a point as one variable( say x)tends to $\mathrm{\infty }$$\underset{x\to \mathrm{\infty }}{lim}$,the value that it approaches (or) converges to ( y value) ,is the value of the asymptote ,the value that the curve tries to reach(y reluctantly tries to get to )but never reaches (or) only reaches the value at $\mathrm{\infty }$.Now the equation of Hyperbola is given by$\frac{y²}{b²}$To find the asymptotes we substitute $\frac{y²}{b²}$Why do we do that ?What is happening here?Is there any geometrical reasoning for this?To find the asymptotes we take the RHS of the equation as 0,why so?

Janiya Rose 2022-08-10

### How to find out whether a function has vertical asymptotesI got the following question wrong in my exameMy answer was, -16,16 for the horizontal asymptotes and 4,-4 for the vertical asymptotes.For the horizontal asymptotes the answer is right, But there aren't any vertical asymptotes.I thought if there wasn't a sum of squares in the denominator There would be vertical asymptotes.So I proceeded this way, as I need to equal the denominator to 0 in order to find the vertical asymptotes.$|x|-4=0$x−4=0x=4−x−4=0−x=4(−1)x=−4There are 3 rules to find the HA's, but what are the rules to find the VA's ?

brasocas6 2022-08-07

### Finding asymptotes of a function using limitsI need to find asymptotes of a function. I didn't find any vertical asymptotes and I think there aren't any. But when trying to find a horizontal asymptote I evaluate the limit as x approaches infinity but can't successfully find it.$f\left(x\right)=x\mathrm{arctan}\left(x\right)$

Braylon Lester 2022-07-23

### Asymptotes of functionHow to find a if the function below doesn't have a vertical asymptotes.$f\left(x\right)=\frac{{x}^{2}-ax+2}{x-2}$

Mbalisikerc 2022-07-22

### Parallel AsymptotesBy definition, any lines that are not parallel will intersect eventually. But to be parallel lines, both lines must have the same slope. However, consider this situation:There is a exponential graph that is an asymptote on the y-axis and on the same graph there is a reflection across the y-axis of the first exponential curve.Now, we know that since these lines are asymptotes they will never actually reach the y-axis (where they would intersect). But also, they do not have the same slope. So - my question is - are these lines parallel?

wstecznyg5 2022-07-21

### Asymptotes In Pre-CalculusI am currently learning about the asymptotes of rational functions in precalculus. However, 4 things about it confuse me, specifically about the asymptotes of it, and how to calculate them.Why is it when given a rational function with 2 polynomials of the same degree on their numerators and denominators, the asymptote can be found by dividing the coefficients of the highest degree terms?Why is it if the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote?Why is it if the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote?Why is it when the polynomial in the numerator is a higher degree than the polynomial in the denominator, there will be a slant asymptote which you can find through polynomial long division?Can you explain all of this using simple algebra, without complicated techniques such as Euler's division of polynomials? I don't understand any complicated techniques and theorems beyond the quadratic formula. Can you also show and explain your working, so it is easier for me to follow through? I am still a beginner, so that would help very much

Luz Stokes 2022-07-19

### Can asymptotes be curved?When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form y=a) and vertical ones ( of form x=b).I was then shown oblique asymptotes-- slanted asymptotes which are not constant (of the form y=ax+b).What happens, though, if we've got a function such as$f\left(x\right)={e}^{x}+\frac{1}{x}?$Is $y={e}^{x}$ considered an asymptote in this example?Another example, just to show you where I'm coming from, is$g\left(x\right)={x}^{2}+\mathrm{sin}\left(x\right)$-- is $y={x}^{2}$ an asymptote in this case?The reason that I ask is that I don't really see the point in defining oblique asymptotes and not curved ones; surely, if we want to know the behaviour of y as $x\to \mathrm{\infty }$, we should include all types of functions as asymptotes.If asymptotes cannot be curves, then why arbitrarily restrict asymptotes to lines?

Jaylene Hunter 2022-07-18

### Find the equation of the horizontal and vertical asymptotesFind the equations of the asymptotes for the following function:$\frac{{x}^{2}+8}{{x}^{2}-9}$My solution is the asymptotes are first to find the vertical asymptotes. To do this, I have to find the value that make expression undefined. As a result, the vertical asymptotes are x=−3 and x=3. To find the horizontal asymptotes, I have the compare the degrees of the numerator and the denominator, which result in y=1My question is, how do I find the equations for the vertical and horizontal asymptotes? Thank you.

Mbalisikerc 2022-07-18

### Find all asymptotes of a functionFind all asymptotes of a function:$f\left(x\right)=\mathrm{log}\left({x}^{2}-4\right)$Domain: $x\in \left(-\mathrm{\infty },-2\right)\cup \left(2,\mathrm{\infty }\right)$Vertical asymptotes are x=−2 (left) and x=2(right):I calculate the limits in +/- infinity:So I'm looking for the oblique asymptotes of a form y=Ax+B:The same for $-\mathrm{\infty }$. How should I interpret this? There are no oblique asymptotes?

Marcelo Mullins 2022-07-17

### Vertical/horizontal asymptotesWe know that for rational functions$f\left(x\right)/g\left(x\right)$vertical asymptotes are defined as the lines $x={x}_{0}$ where $g\left({x}_{0}\right)=0$, and for horizontal asymptotes it is the limit as x approaches $\mathrm{\infty }$ or x approaches $-\mathrm{\infty }$. My question is the following: what is the relationship between vertical and horizontal asymptotes? For instance, if we have$y=f\left(x\right)$and we express x as function of y, or$x={f}^{-1}\left(y\right),$is the horizontal asymptote of inverse function the same as the vertical one of the original?Thanks in advance.

glyperezrl 2022-07-17