Question

Could someone please help and explain to me:
Given lim x->0 (1+2x)^(csc(x)) = e^2
Use the limit properties to determine the value of lim x->0 ((1+2x)^(csc(x)))/(3x^2-4x+5) ?

Answer 1

This isn't incredibly rigorous. I know the answer is right, but that might be an accident. Anyways, this might be some help.

What's the lim x -.> 0 1/(3x^2-4x+5) >?

Answer 2

1/5?

Answer 3

yah

Answer 4

and I'm pretty sure this is valid

\[\lim_{x \rightarrow 0}\frac{(1+2x)^{\csc (x)}}{(3x^2-4x+5)} = \lim_{x \rightarrow 0}\frac{1}{(3x^2-4x+5)}\lim_{x \rightarrow 0}(1+2x)^{\csc (x)}\]

Answer 5

It gives the right answer in this case, but it might be a taped together poorly...

Answer 6

so 1/5 * e^2 is the answer?

Answer 7

yeah, e^2 / 5

Answer 8

okay, thank you so much! :) :D

Answer 9

:) welcome. just be wary of using that other places until someone with more knowledge says when it's valid and not valid! :P

Answer 10

Yeah, that definitely isn't true in most cases, fyi. But the answer's right nonetheless :/

Answer 11

sorry, but I realized I made a slight typo previously,
(1+2x)^(2cscx) instead of (1+2x)^(cscx) in the determining the value part... how would that change the answer?

Answer 12

It makes it e^4 instead of e^2, but I'm not sure how to show it. Using the same logic as before, (which is !!VERY!! spotty at very best)

e^4 = e^2*e^2 , just as (1+2x)^(cscx) = (1+2x)^(cscx) * (1+2x)^(cscx)

That pattern holds true for other coefficients in front of the csc, but again, that might just be a coincidence.

Answer 13

But flash this question at one of the moderators / higher ups to make sure any of that (the answer is right, I know, but...) is correct :)

Answer 14

okay, thank you :)

Answer 15

and this should be

e^4 = e^2*e^2 , just as (1+2x)^(2cscx) = (1+2x)^(cscx) * (1+2x)^(cscx)