Question

what is an oblique asymptote?

Answer 1

It's when you have an asymptote that's not parallel to the x- or y-axis.

Answer 2

The blue line's the oblique asymptote.

Answer 3

The other asymptote, x = 0 (i.e. the y-axis) is not oblique.

Answer 4

so when would you have one?

Answer 5

I'm just looking for a function/./.

Answer 6

You know, instead of me repeating everything, there's some useful information here:

http://en.wikipedia.org/wiki/Asymptote

Answer 7

You basically have an oblique asymptote if,\[\lim_{x \rightarrow \infty}[f(x)-(mx+b)]=0\]where f is your function and mx+b is a straight line.

Answer 8

That's the definition.

Answer 9

Are you okay with that or do you need to run through an example?

Answer 10

may i please have an example. thankz

Answer 11

my teacher said that we know there is an OA when the question is a fraction and the power of the numerator is greater than the power of the denominator or something like that

Answer 12

but i was a bit confused about that

Answer 13

Okay. You could have a function like\[y=x+\frac{1}{x}= \frac{x^2+1}{x}\]

Answer 14

From the definition, we check for the slope of the OA:\[m:=\lim_{x \rightarrow \pm \infty}\frac{f(x)}{x}=\lim_{x \rightarrow \pm \infty}\frac{(x+\frac{1}{x})}{x}=\lim_{x \rightarrow \pm \infty}(1+\frac{1}{x^2})=1\]

Answer 15

so that's your slope

Answer 16

Now you need to check for your y-intercept:

Answer 17

\[n:=\lim_{x \rightarrow \pm \infty}(f(x)-mx)=\lim_{x \rightarrow \pm \infty}((x+\frac{1}{x})-x)=\lim_{x \rightarrow \pm \infty}(\frac{1}{x})=0\]

Answer 18

(should read 0 at end).

So the equation of your OA is\[y=x\]

Answer 19

Both the limits have to exist for an OA.

Answer 20

in your working out how did you get lim[1+(1/x^2)]

Answer 21

how did you conclude that y=x?

Answer 22

Because\[\frac{x+\frac{1}{x}}{x}=\frac{x}{x}+ \frac{\frac{1}{x}}{x}=1+\frac{1}{x} \times \frac{1}{x}=1+\frac{1}{x^2}\]

Answer 23

y=x because m=1 and n = 0.

Answer 24

The process is all about finding a line, y=mx+n (if it exists) that will be an OA.

Answer 25

oh ok i think i should go and do some more practice egs
thanks for writing all the working out! you make a brilliant teacher!

Answer 26

you're welcome :)

Answer 27

oh umm do you know any websites that have practice questions on differentiation curve sketching?

Answer 28

Not off the top of my head. I can have a quick look.

Answer 29

Is this what you're looking for?

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/graphingdirectory/Graphing.html

Answer 30

Just do a Google search for "curve sketching problems".

Answer 31

ahhh thank you very much for the site

Answer 32

np