how to tell if asymptote at y=x (1 Viewer)

[TheGreatest99.95]

so basically with some of the mx2 graph questions they have an asymptote at y=x.
How can you test to see if there is an asymptote at y=x? i tried asking my teacher and she didnt make sense.
Sorry if it seems like a noob question, never been taught it

Cheers and goodluck on monday

 

[Trebla]

If there is an asymptote of y = x, then when x gets large to infinity (similarly for minus infinity) y approaches infinity but does so at the same rate that x goes to infinity. As an example, consider y = x + 1/x. As x gets large, 1/x approaches zero but the first term approaches infinity, which means that y approaches infinity at the same rate that x approaches infinity thus an asymptote exists at y = x.

Another possible way to detect if there is a y = x asymptote is to evaluate y/x and see if it approaches 1 as x approaches infinity. So in this example, y/x = 1 + 1/x². So when x approaches infinity, y/x approaches 1.

 

[D94]

This should be obvious by now... generally for MX2, you don't need calculus to draw a curve (whereas in MX1, you may be asked to find the stationary points by differentiation etc.).

There's usually 6 things you need to test for:
1. Vertical Asymptote
2. Horizontal Asymptote
3. Oblique Asymptote
4. Intercepts
5. Nature of curve as it approaches the Vertical Asymptotes on both positive and negative sides
6. Check if the curve cuts the Horizontal Asymptote

So the y = x asymptote would go under the Oblique Asymptote test. To find it, you would either need to use polynomial long division, manipulating the equation to separate terms, or by observation for more simpler equations.

Also, the oblique/slant asymptote could be a parabola or it could be a hyperbola. It doesn't have to be y = x.