Question

ƒ(x)=6x-5/x-3

List all horizontal asymptotes of ƒ. If there are no horizontal asymptotes, enter 'NONE'.
Horizontal asymptotes y?

Answer 1

Let f and g be polynomials
\[\lim_{x \rightarrow \infty } \frac{f(x)}{g(x)}\]

*If deg(f)=deg(g), then the horizontal asymptote is y= coefficient of the term with the biggest exponent on top/ coefficient of the term with the biggest exponent on bottom

*If deg(f)>deg(g), then no horizzontal asymptote

*If deg(f)<deg(g), then y=0 is the horizontal asymptote

Answer 2

Where deg( ) means degree of

Answer 3

So both your f and g are polynomials
and what can you say about there degrees?

Answer 4

Is this a trick ?

Answer 5

so what's the answer? 3?

Answer 6

No

Answer 7

for your f what is deg(f)?
for your g what is deg(g)?
After you decide this, determine what if part above your problem satisfies to find the conclusion to your problem.

Answer 8

Thanks so clear now, the answer is 6! hehe

Answer 9

If the distance between the graph of a function and some fixed line approaches Zero as point on the graph moves increasingly far from the origin, we say that the graph approaches a line asymptotically and the line is an asymptote of the graph.

For horizontal asymptote, a line \( y=b\) is horizontal asymptote of the graph of a function \( y=f(x) \) if either \( \lim \limits_ {x\to \infty } f(x)= b\) or \( \lim \limits_ {x\to -\infty } f(x)= b\)

Answer 10

So, just take the limit to infinity, what Myin's gave you is simpler and faster way to detect the asymptote.

Answer 11

Thanks FoolForMath!!

Answer 12

Another cute thing to note is that

"*If deg(f)>deg(g), then no horizzontal asymptote" but there will be oblique asymptote.