finding limits

Hello guys,

is there a set method on finding the limits of more complex functions like ones with "e"s in the numerator and denominators, logs n such? I seem to remember vaguely a few years back my teacher mentioning something about finding whichever (denominator or numerator) dominates by finding the dominant terms, then go from there, like if one of them dominates then the limit is 0 or something...(of course this is all with Limit->infinity, I assume with finite limits you just plug in the number or factor as appropriate)

Any tips on how to solve for complex limits are welcomed!!
Derg

Quote:

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Originally Posted by Dergyll

Hello guys,

is there a set method on finding the limits of more complex functions like ones with "e"s in the numerator and denominators, logs n such? I seem to remember vaguely a few years back my teacher mentioning something about finding whichever (denominator or numerator) dominates by finding the dominant terms, then go from there, like if one of them dominates then the limit is 0 or something...(of course this is all with Limit->infinity, I assume with finite limits you just plug in the number or factor as appropriate)

Any tips on how to solve for complex limits are welcomed!!
Derg

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it would be best if you come up with specific problems here. it would be unreasonable to go through all the methods here. that is something you should consult a calculus book or an online resource for.

ok, lets say:

(e^2)(4x^2)200Log(20) / (3e^0.5x)x^3

how would I solve something like this?

sorry, the log is base 10 (does it matter?) and the Limit is as X->infinity

the limit as x goes to?

Ok guys, how about this one (don't tell me to graph it then find the limit please, calculation only)

(x^4) + (5x^2) + 1 limit as x->infinity

Thank you!

Quote:

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Originally Posted by Dergyll

Ok guys, how about this one (don't tell me to graph it then find the limit please, calculation only)

(x^4) + (5x^2) + 1 limit as x->infinity

Thank you!

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It's infinity by any definition I can think. To prove it merely note that given any $\displaystyle T>0$ we can make $\displaystyle x^4+5x^2+1>T$ by choosing $\displaystyle x>T$

this is a pretty simple limit:

$\displaystyle x^4 + 5x^2 +1$, if you take it to $\displaystyle \infty$
then all the x's will go $\displaystyle \infty$ and the entire expression
will go to $\displaystyle \infty$.

a tricky limit will be one where you can't really just substitute the
variable with $\displaystyle \infty$ or the variable is also in the
denominator and it goes to 0...

I found this on wikipedia!!!

"There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x):

If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
If the degree of p is less than the degree of q, the limit is 0.
If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; they may occur with rational functions."

Of course this is only for fractions, but it helps alot! Can I always use the method where you substitute Infinity into the equations? Won't the equations not make any sense?

Derg