Question

Limits,
http://screencast.com/t/MeamUq9EgG8h

I heard there is a rule about "powers" when it comes to limits like this in order to solve them? Can you tell me the rule please?

Answer 1

Is this your question ?\[\large\lim_{x\to+\infty}\frac{3x+1}{2x-5}\]

Answer 2

yes

Answer 3

Yes there is a nice little shortcut, which is what you must be referring to.
I'm trying to think of how to explain it.

Since it's an infinite limit, we only need to worry about the `leading` term (highest power on top and bottom).

Compare them, if they're equal degree, the limit will approach the ratio of their coefficients. In this case, 3/2.

Answer 4

when \(x\to+\infty\), the ratio will ok like \(\frac{3x}{2x}\)

Answer 5

you mean like 3/2 ??

Answer 6

and when x->-inf -3/2 ?

Answer 7

Both x's are approaching negative infinity, so you can think of the negatives cancelling each other out, giving us the same 3/2 as with before :)

Answer 8

What about higher power at the denominator or the numerator separately

Answer 9

@zepdrix ?

Answer 10

It might help if you see an example and just think about it logically.

\[\large \lim_{x \rightarrow \infty}\frac{\color{green}{x^2-2x+3}}{\color{royalblue}{x-7}}\]

The green part represents a parabola,
while the blue part a line.

Which one will grow towards infinity faster?
A straight line or a parabola?

Answer 11

1 ?

Answer 12

what? XD lol

Answer 13

uhm I think its the whole numerator? I am not sure :D

Answer 14

noooo it's not :O
See how the `power` on x is larger in the top?
We can't apply the same rule we used before, since the highest powers of x `do not match`.

Answer 15

Which one of these functions is growing upwards faster? :O