how do you find the limit of (1-3/x)^2x as x approaches negative infinity?

Question

i looked at a previous example which had the limit of (1/1/2x)^3x as x approaches positive infinity. You make 2x equal to some variable u so x=u/2. so it would be the limit (1+1/u)^3(u/2). You put the u as a power of (1+1/u), which gives you e to the 3/2 power as the answer.

So, I'm trying to use that principle to apply here. Could someone help? Thanks

Answer 1

We have this fact:

lim as x→∞ (1 + r/x)^x = e^r

Let x = -y . This limit becomes

lim as y →-∞ (1 - r/y)^(-y) = e^r

Inverting both sides,

lim as y →-∞ (1 - r/y)^y = e^(-r)

Let y = 2u

lim as u →-∞ (1 - r/(2u))^2u = e^(-r)

Write this as

lim as u →-∞ (1 - (r/2)/u)^2u = e^(-r)

Let s = r/2

lim as u →-∞ (1 - s/u)^2u = e^(-2s)

Now take s=3

Answer 2

Let -3/x = 1/t. Thus, x = -3t => 2x = -6t
limit of (1 - 3/x)^2x , as x->-∞
= limit of (1 + 1/t)^(-6t), as t->∞
= e^-6
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Ideas to save time: Use substitution -3/x = 1/t so that the base becomes 1 + 1/t

Answer 3

first difficulty is to submit to in suggestions prevalent shrink regulations do no longer practice to applications dealing with infinity. So what to do then? you opt for for look on the function and locate the utmost skill of x. (x^2) Then divide the completed function with the aid of the utmost skill So we could get: (one million/x) -------------------- (sixteen/x^2) + one million ok now we are able to cancel out (one million/x) and (sixteen/x^2) by using fact if we plug in an somewhat large unfavorable selection (ex. -one million billion) for x for (one million/x) we could get -.000000001 which rounds to 0 and we would get an identical answer for (sixteen/x^2). so as that leaves us with: 0/0+one million = 0 meaning as x gets infinitely greater in a unfavorable direction f(x) gets infinitely nearer and nearer to 0. the adaptation between this and the previous strategies is the quantity of artwork you need to instruct. some instructors will require you to instruct no longer basically the logic however the algebra as properly. examine with your instructor and notice with approach his or she would manage to settle for. i did no longer positioned the algebra on considered one of my attempt and the instructor docked me 2 or 3 factors for the undertaking.