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Asymptotes

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Asymptotes, a selection of answers from the Dr. Math archives.

Etymology of 'Asymptote'
Where does the term 'asymptote' come from?

Asymptotes
When do asymptotes occur?

Vertical and Horizontal Asymptotes
What are the vertical and horizontal asymptotes?

Graphing a Function with Asymptotes
How do you find the asymptotes of f(x) = (2x + 1)/(x - 3)? How do you use the asymptotes to graph the function?

Finding Asymptotes
In the equation y = (m/x) + c, how do the values of m and c affect the graph? How do you find the asymptotes?

Domain, Range, and Asymptote
How do you find the domain, range, and asymptote of a function?

Asymptote of a Function
Determine the value of A so that y = (Ax+5)/(3-6x) has a horizontal asymptote at y = -2/3.

Testing for Horizontal Asymptotes
Is there a rule for testing whether or not an equation has a horizontal asymptote?

Finding a Vertical Asymptote
Find the vertical asymptote of the equation xy^2 - x^3y = 6.

Domain, Asymptotes, Intercepts of a Function
What is the domain of this function? What asymptotes does it have? What are the x and y intercepts? Etc...

Oblique Asymptotes
I'm stuck on how to calculate oblique limits, for example, x^2/(x-3).

Oblique Asymptote
I have a question about the oblique asymptote of the function: F(X)=(X^3-3X^2+2X-8)/(X^2+4X+4)...

Nonvertical Asymptotes
Does this problem have slant asymptotes?

Line Crossing Horizontal and Vertical Asymptotes
For the function y = (5x+1)/(x^2-1), why does the line between its two vertical asymptotes and one horizontal asymptote cross?

Rational Function
Given the rational function y=r(x)=(4x^6-x^4)/(x^5+5) describe its end behavior and find all vertical and horizontal asymptotes for r if they exist.

Similar Graphs
How would you describe how the two graphs of y = 1/sqrt(x)^2 and y = 1/sqrt(x)^2-4 are similar and dissimilar?

Trigonometric Functions and the Unit Circle
Why don't sine and cosine graphs have values greater than 1 or less than -1? Why does the tangent graph have asymptotes? How do the trig functions relate to the unit circle?

Analyzing a Function
Find the domain, intercepts, asymptotes, critical points, points of inflection, and graph of the function: y = 3x^4-4x^3-12x^2.

Dominant Terms
What are dominant terms, and how do you obtain their values?

Focus of a Hyperbola
When you rotate the focus point of a hyperbola around the center, why does it hit the asymptote with the same x coordinate as the vertex?

X = TanX
Show that x = tanx has an infinite number of solutions.

Asymptote for (x-4)/x^2
For the function (x-4)/x^2 there is a horizontal asymptote at y=0; however, at x=4 y=0 the graph of the function does intersect the line that is supposed to be the asymptote. How can this be if by definition the asymptote is a line that the function gets close to but does not touch?

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