Christos
  • Christos
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?
Mathematics
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Christos
  • Christos
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?
Mathematics
schrodinger
  • schrodinger
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UnkleRhaukus
  • UnkleRhaukus
Is this your question ?\[\large\lim_{x\to+\infty}\frac{3x+1}{2x-5}\]
Christos
  • Christos
yes
zepdrix
  • zepdrix
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.

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UnkleRhaukus
  • UnkleRhaukus
when \(x\to+\infty\), the ratio will ok like \(\frac{3x}{2x}\)
Christos
  • Christos
you mean like 3/2 ??
Christos
  • Christos
and when x->-inf -3/2 ?
zepdrix
  • zepdrix
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 :)
Christos
  • Christos
What about higher power at the denominator or the numerator separately
Christos
  • Christos
zepdrix
  • zepdrix
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?
Christos
  • Christos
1 ?
zepdrix
  • zepdrix
what? XD lol
Christos
  • Christos
uhm I think its the whole numerator? I am not sure :D
zepdrix
  • zepdrix
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`.
zepdrix
  • zepdrix
|dw:1369665937413:dw|Which one of these functions is growing upwards faster? :O
zepdrix
  • zepdrix
The curve or the straight line? :D
Christos
  • Christos
numerator = denominator?
Christos
  • Christos
Eeeh i mean that high one not the straight one the other one grows faster
zepdrix
  • zepdrix
Yes good good, the curvy one grows faster.
zepdrix
  • zepdrix
\[\large \lim_{x \rightarrow \infty}\frac{\color{green}{x^2-2x+3}}{\color{royalblue}{x-7}}\]The `green` part is THAT type of function, the curvy one. So we can think of it this way, as the numerator and denominator approach infinity, the numerator is "winning", it's getting there faster. So the limit will approach infinity.
zepdrix
  • zepdrix
That too confusing? :3 stick with the math rules?
Christos
  • Christos
same applies for x->-inf ? and what about the denominator having the highest power?
zepdrix
  • zepdrix
Let's look at the reverse a moment. \[\large \lim_{x \rightarrow \infty}\frac{\color{royalblue}{x-7}}{\color{green}{x^2-2x+3}}\] Now the BOTTOM is a larger degree than the top. So the bottom is "winning", approaching infinity faster. So our limit will approach \(\large \dfrac{1}{\infty}\) which is the same as saying, it's approaching zero. The denominator is getting bigger and bigger faster than the top, so the fraction becomes smaller and smaller.
Christos
  • Christos
thanks
zepdrix
  • zepdrix
Negative infinity is a little trickier, since even powers will mess things up sometimes... I'm not sure how to explain that part properly :3
Christos
  • Christos
Just got back from my final exams on Calculus 1, 95% + :D .. I still can't believe that 3 months earlier I couldn't even do algebra. I got submitted to the University 3 weeks after the 1st semester was taking place.. They said no way a student with 10/20 overall score in high-school Algebra can take Calculus 1 from the very first semester, starting the course 3 WEEKS later.. I took this as a challenge, studied Algebra for additional 3 weeks straight. 6 weeks had past since the course of Calc had started at the uni. I went to the advisor's office, and said "I want to start Calculus 1. Give me the admission's test, if I pass, you let me in" . It was an algebra based test, scored 25/30 on it. Qualifying score to start studying Calculus 1 immediately. Teachers still doubting me, my score was pretty high on algebra they said, but starting Calculus 1 6 weeks after the lessons were taking place, I had 10% chance of even PASSING the course, they said. And here I am today, with OpenStudy as my friend and teacher, during this 3 months I studied algebra , qualified to take one of the most advanced forms of math Calculus 1 , scored 95%+ One of the highest if not the highest score in my class. Never in my life I thought that there would come a point where I would say "I love Math". A very big Thank You to the OpenStudy Community.
reemii
  • reemii
Congrats!
UnkleRhaukus
  • UnkleRhaukus
Well done @Christos ! \(\LARGE\mathcal{CONGRATULATIONS!!}\)

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