Maths Ext 1 - Sketching/Limits Question (1 Viewer)

[iPhone4]

Hi everyone,

I got the question below out of our textbook (Maths In Focus - Preliminary Ext 1 - Exercise 5:11 Q5f)

"Sketch"

y = 1 + x
x^2+1

Do it in a book or on a piece of paper.

The line seems to go through 0, 1 but the problem is, is that it approaches 1 but never touches it later on.
If you sketch it, you will see what I mean.

My question is: Is the line y=1 a horizontal asymptote? Can lines go through asymptotes?

Thanks

 

[funnytomato]

yes
yes

 

[funnytomato]

remember the function asymptotically approaches y=1 as x tends to +ve/-ve infinity
that does not show us any infomation about its behaviour when x is finite, so it could cross it at some finite value

does that make sense?

 

[Spiritual Being]

I'm going to assume you mean y=(1+x)/(x^2+1). In that case, y=1 would not be a horizontal asymptote. To find the horizontal asymptotes you must first look at the degrees of the numerator and the denominator. If the degree of the denominator is larger than the degree of the numerator, than the horizontal asymptote is at y=0, as is the case here. (If the degree of the denominator=degree of the numerator, the horizontal asymptote is the leading coefficient of the numerator over the leading coefficient of the denominator, and if the degree of the denominator<degree of the numerator, there is no horizontal asymptote). And yes, the graphs of equations can go through HORIZONTAL asymptotes (not vertical asymptotes). In fact, this graph does. You know because if you set y=0 and solve, you'll get x=-1. Thus, the graph crosses the horizontal asymptote at x=-1 to make the point (-1,0). An asymptote is just the value that a function approaches as it approaches positive or negative infinity.

If you think about it, all real numbers must make the denominator positive, and all numbers<-1 will make the numerator negative, so all values of x<-1 will make the value negatiive, at -1 it will cross the x-axis/asymptote, and all values of x>-1 will make the value positive. So as x approaches positive infinity, it approaches the asymptote from "above" or from a positive value, and as x approaches negative infinity, it approaches the asymptote from "below" or from a negative value.




hope you like my general maths skills ppppppp

 

[iPhone4]

Oops! The way I wrote the formula didn't come across properly.


It is y= 1 + (x/x^2+1)

 

[RivalryofTroll]

Oops! The way I wrote the formula didn't come across properly.


It is y= 1 + (x/x^2+1)

y = 1 is an asymptote.
Lines CAN go through the horizontal asymptote but not the vertical asymptotes.

 

[iPhone4]

Well it begs the question - is it really an asymptote?

When you substitute Y=1 into the formula, you get X=0.

I wrote this in my book:

When x>0, y(cannot equal)1
When x<0, y(cannot equal)1
When x=0, y=0

Then drew the graph, but didn't mark y=1 as an asymptote as the line travels through it.

 

[deswa1]

The definition of an asymptote is "a line such that the distance between the curve and the line approaches zero as they tend to infinity"- From wiki. It doesn't matter how many times the curve crosses the asymptote, you can even draw curves that cross an asymptote an infinite number of times.

 

[RealiseNothing]

It still approaches y=1 as x tends to infinity(+ and -) though.

 

[deswa1]

Which is why y=1 is a horizontal asymptote.

 

[RealiseNothing]

Which is why y=1 is a horizontal asymptote.

Was talking to OP lol.

 

[iPhone4]

Thanks guys/girls - I wasn't aware that lines could travel through asymptotes without being discontinuous.

 

[nightweaver066]

Thanks guys/girls - I wasn't aware that lines could travel through asymptotes without being discontinuous.

Only asymptotes that are not vertical.