hi people,
i a maths learner and i need help with this limits question i dont know how to answer because of the letters, can someone please answer it.
hi people,
i a maths learner and i need help with this limits question i dont know how to answer because of the letters, can someone please answer it.
$\displaystyle \lim_{x\to\infty} \frac{a_1x^2+ b_1x+ c_1}{a_2x^2+ b_2x+ c_2}$?
General rule for limits of rational functions (a polynomial divided by a polynomial) as x goes to infinity: divide both numerator and denominator by the highest power of x.
Here, $\displaystyle \frac{a_1x^2+ b_1x+ c_1}{a_2x^2+ b_2x+ c_2}= \frac{a_1+ \frac{b_1}{x}+ \frac{c_1}{x^2}}{a_2+ \frac{b_2}{x}+ \frac{c_2}{x^2}}$.
Now, what happens to the "little" fractions as x goes to infinity?
$\displaystyle \lim_{x \rightarrow \infty} \frac{a_1x^2+b_1x+c_1}{a_2x^2+b_2x+c_2} = \lim_{x \rightarrow \infty}\frac{x^2(a_1+\frac{b_1}{x}+\frac{c_1}{x^2} )}{x^2(a_2+\frac{b_2}{x}+\frac{c_2}{x^2})}$
Now 'cut' those $\displaystyle x^2$ off and find limit of numerator and denominator of the fraction.
Regards.
All the question says is, evaluate the following, so i dont know what its realy asking for im new at limits, but the answer from the back of the sheet to the question is a1/a2
im guesing its a problem that is done with just canceling out??
Since you have been given a number of responses, why are you still guessing?
pOoint and I said essentially the same thing. Now, do you know what $\displaystyle \lim_{x\to\infty} \frac{1}{x}$ and $\displaystyle \lim_{x\to\infty} \frac{1}{x^2}$ are?
If not, what is your understanding of what a "limit" is?
yeh im still asking because the answers i have gotten do not match the right answer which is a1/a2, and i have a very small knowledge of limits, trying to learn the infiity limits now, and HallsofIvy is the answer undefined? 1/x well would equal 0 because if you try larger variables for x it gets closer to 0. im just a n00b and i just wanted some advice to make limits easier for me thats all, i still dont understand that question
No. The answer is 0. This is something fundamental. If you do not know it then (and I'm sorry for any offence this will cause) you have no business attempting the question you have posted until you do know it. It's pointless attempting questions that rely on a non-existent background. Go back, review the basics and then attempt questions like this one.
(The answer of a1/a2 follows directly from the two replies given to you by HoH).
When mr fantastic says "The answer is 0", he is referring to my questions about "lim 1/x" and "lim 1/x^2". You said that "1/x well would equal 0 because if you try larger variables for x it gets closer to 0." which is correct and the same thing is true for 1/x^2. But you shouldn't need to " try larger variables for x". It should be clear that if the numerator is always 1 and the denominator gets larger and larger, the entire fraction goes to 0.
You need to know the basic "laws of limits":
If $\displaystyle \lim_{x\to a} f(x)= L$ and $\displaystyle \lim_{x\to a} g(x)= M$ then
a) $\displaystyle \lim_{x\to a} f(x)+ g(x)= L+ M$
b) $\displaystyle \lim_{x\to a} f(x)g(x)= LM$
c) $\displaystyle \lim_{x\to a} f(x)/g(x)= L/M$ as long as $\displaystyle M\ne 0$.
Now, that means that $\displaystyle \lim_{x\to\infty}\frac{a_1x^2+ b_1x+ c_1}{a_2x^2+ b_2x+ c_2}= \lim_{x\to\infty}\frac{a_1+ \frac{b_1}{x}+ \frac{c_1}{x^2}}{a_2+ \frac{b_2}{x}+ \frac{c_2}{x^2}}$$\displaystyle = \frac{\lim_x\to\infty} a_1+ b_1\lim_{x\to\infty}\frac{1}{x}+ c_1\lim_{x\to\infty}\frac{1}{x^2}}{\lim_{x\to\inft y}a_2+ b_2\lim_{x\to\infty}\frac{1}{x}+ c_2\lim_{x\to\infty}\frac{1}{x^2}}$