Find the limit as x approaches a constant

[Lo.Lee.Ta.]

1. a. lim (-15)/((x^3)(x + 8)^4) = ?
x→-8-

b. lim (-15)/((x^3)(x + 8)^4) = ?
x→-8+

c. lim (-15)/((x^3)(x + 8)^4) = ?
x→-8


2. I don't think I've ever done a problem like this before...
I saw on one website (http://openstudy.com/updates/4f7761e7e4b0ddcbb89dced0)
that this person advised to replace the x→-8- with h→0.
He/she said to make x = -8 - h.

So, taking this advice, my problem is:

lim (-15)/[(8 + h)^3(-8 - h + 8)^4]
h→0

= lim (-15)/(-h^3 - 112h - 3h^2 -576)(-h)
h→0

Then I thought that I would replace the h's with 0's.

(-15)/(0 - 0 - 0 - 576)(0)

=-15/0 = Does not exist? :/

This is not right, I don't think.
Please help me to figure out this problem!
Thank you SO much! :)

 

[Joffan]

The idea is to find out what happens to the function when x is just less than -8 as it gets closer, and (second part) when it's just more. and getting closer to -8. Then if they match, that's the limit value (third part), or if they don't match there's no limit value.

For most functions we use, at most values, the limit from both sides is the same as the function value at that point (roughly the definition of continuous). So it is only usually interesting when the function at that value is behaving oddly.

Typically you'll use δ to indicate the difference of x from the target value, so in the first case you'll examine the function for x = -8-δ (in the knowledge that δ>0 is very small), and see what happens as δ→0. You've started to do this (using h) although perhaps with more detail than you really need.

 

[symbolipoint]

After trying some numeric manipulations and not achieving anything, I tried looking carefully at the expressions that constitute the full expression to be evaluated.

Approaching -8 from either side, you see that you will have (-)/((-)(+)), so whatever happens close to or at -8, the value will be positive or nonexistent possibly.
Looking at factors of the denominator, (x+8)^4 will get smaller faster than x^3 will get larger in size. The denominator then seems to decrease without bound, making the entire rational expression INCREASE without bound, except that the expression has no value at x=-8.

Conclusion is that analysis is fair, limit does not exist.

 

[haruspex]

= lim (-15)/(-h^3 - 112h - 3h^2 -576)(-h)
h→0
Then I thought that I would replace the h's with 0's.

The trick at this point is to set zero all those h's that can't be making any difference, such as those in (-h^3 - 112h - 3h^2 -576). Clearly that part will tend to -576.

 

[Lo.Lee.Ta.]

So this method I am using is right?

@haruspex - You say that the h's here (-h^3 - 112h - 3h^2 -576)
won't matter.
But the (-h) does matter? And you don't replace that one with zero?

But how to you know which h's will matter...?


lim (-15)/(-h^3 - 112h - 3h^2 -576)(-h)
h→0

So it would end up as: -15/(-576)(-h)

How would I solve it from there? There's still a -h left...
Thanks for your help!

 

[haruspex]

But how to you know which h's will matter...?

Clearly the expression (-h^3 - 112h - 3h^2 -576) tends to -576 as h tends to zero. Since that is neither 0 nor infinity, and is a factor of the whole expression, you can safely factor that out and take its limit independently.

So it would end up as: -15/(-576)(-h)
How would I solve it from there? There's still a -h left...

Yes, but you can simplify the signs and factors to produce -(5/192)(1/h). What do you think that will do as h approaches zero from the negative side?

 

[Joffan]

What happened to the fourth power? and 8^3=512.

 

[Lo.Lee.Ta.]

@Joffan - Thanks for pointing that out! I completely cubed the (-8 -h)^3 wrong!

I think it should be: -h^3 - 24h^2 - 192h - 512

If zero replaces the h's, it's just -512.

So it's: -15/(-512 * -h)

@haruspex - So you want me to consider what the fuction equals as h (or x) approaches 0 from the left?
Approaching 0 from the left, I come across numbers like -5, -2, -1, -.5...
x=-5, y=.00586

x=-3, y=.0146

x=-1, y=.0293

x=-.5, y=.0586

x=-.01, y=2.93

x=-.00000000001, y=2929687500

So when x gets closer to zero, y increases. The line is getting taller as x gets grows.

So it looks like y approaches +infinity.
Is that right?
Is the answer that the function = +infinity?And for the part of the question asking about what the function equals as x approaches -8 from the right- is the answer to that -infinity?
If I use this previous reasoning (don't know if it's right, though... :/), the answer seems like it would be -infinity...

But I think the computer homework system counted both of these answers wrong when I tried them... :/

 

[haruspex]

x=-.00000000001, y=2929687500

So when x gets closer to zero, y increases. The line is getting taller as x gets grows.

So it looks like y approaches +infinity.

Yes.

And for the part of the question asking about what the function equals as x approaches -8 from the right- is the answer to that -infinity?

Yes again.
So what would you answer for (c)?

 

[Lo.Lee.Ta.]

Thanks for responding, haruspex! :)

Well, when it asks what the function is when x approaches 0, wouldn't it be: -15/(-512*0)

=-15/0 = undefined = limit does not exist? :/

I'm not too sure about this one, but I'm thinking the limit does not exist when x approaches zero...

 

[symbolipoint]

Thanks for responding, haruspex! :)

Well, when it asks what the function is when x approaches 0, wouldn't it be: -15/(-512*0)

=-15/0 = undefined = limit does not exist? :/

I'm not too sure about this one, but I'm thinking the limit does not exist when x approaches zero...

I used a careful qualitative approach, and my conclusion agrees well with yours. You could recheck what I posted and then feel more sure about yourself.

 

[Joffan]

If your function is
$$ \frac{-15}{x^3(x + 8)^4} $$
then I ask again: what happened to that exponent of 4? It makes a difference to one of your answers.

 

[Lo.Lee.Ta.]

@Joffan - You're right! I left that out! :/ Thanks!

I think it should be:

-15/(-512 * (-h)^4)

As h (0r x) approaches 0 from the left, I come across -5, -1, -.001, -.00000000001, etc.

x=-5, y= .000046875

x=-1, y= .029297

x=-.001, y= 29296875000

x=-.00000000001, y= 2.93E42

So as x approaches 0 from the left, y dramatically gets taller.
So I think still that the function equals +infinity.


Approaching 0 from the RIGHT, I come across 5, 1, .001, .00000000001, etc.

x=.029, y=.000046875

x=1, y=.029297

x=.001, y=29296975000

x=.00000000001, y=2.93x10^42

So when x approaches 0 from the right, it looks like y gets dramatically taller.
It's just the same as when we approach from the left.
So is the answer that the function = +infinity?


For the question that asks about what the function equals as x approaches 0 (NOT from left or right), wouldn't we still say that the limit does not exist?

Because -15/(-512 * (0)^4) = -15/0 = undefined = limit does not exist?

Thanks, you guys, for helping me! :)

 

[haruspex]

For the question that asks about what the function equals as x approaches 0 (NOT from left or right), wouldn't we still say that the limit does not exist?

Had the left and right limits been different, you would certainly be right to say the undirected limit is undefined. But since you found both left and right limits to be + infinity, you can declare the undirected limit to be the same.

 

[Lo.Lee.Ta.]

Oh, okay! Everything = +infinity! So the computer counted that as correct!

Thanks SO much haruspex and Joffan and symbolipoint! :D You guys are awesome! :)