Question

Use graphs and tables to find the limit and identify any vertical asymptotes of limit of (1 divided by the quantity x minus 7 squared) as x approaches 7.

Answer 1

If it helps, I'm only confused about the fact that it is approaching 7 and that there's a square in the bottom. I got a similar question but it was [1/(x-9)] as the limit approaches 9 from the right, and I found it quite easy. I am not sure where to begin or how to assess the question.

Answer 2

@Hero could you help me, this is my first question and I'm kind of confused :/

Answer 3

@imqwerty any help at all would be great :)

Answer 4

\[\large \lim_{x\to7}\frac{1}{(x-7)^2}\]In the other problem you dealt with,
\[\large \lim_{x\to9^+}\frac{1}{x-9}\]Notice that the `approaching from the right` is actually really important here!
If we approached from the left side of 9,
then (smaller than 9) - 9 = negative number,
while from the right gives us the opposite sign.

Which is telling us that this limit is undefined,\[\large\rm \lim_{x\to9}\frac{1}{x-9}\]

Answer 5

So what's neat about the square is that it actually deals with this problem!
Whether we're getting a negative or positive in the denominator, it get's squared away, ya?

Answer 6

Let's umm... let's deal with the limit like this...
We know that the denominator is getting closer and closer to zero, right?

So let's plug a value into the limit that's close to zero,
but we'll do so in fractional form.

So let's say the denominator is really really close to zero like
\(\large\rm 0.00001=\frac{1}{99999}\)

Answer 7

Okay, yes I follow so far

Answer 8

\[\large \lim_{x\to7}\frac{1}{(x-7)^2}\quad\approx\quad \frac{1}{\left(\frac{1}{99999}\right)^2}\]

Answer 9

okay

Answer 10

Doesn't that make the number even smaller? And, it also has to do with the negative/positive infinity, right? At least, that's how my teacher was presenting it.

Answer 11

Oh let's say that our tiny tiny fraction can be positive or negative,
just so this takes care of approaching from the left or right, all in one go.\[\large \lim_{x\to7}\frac{1}{(x-7)^2}\quad\approx\quad \frac{1}{\left(\frac{1}{\pm99999}\right)^2}=\frac{1}{\left(\frac{1}{99999^2}\right)}\]The plus/minus gets squared away.

Remember how to deal with division of a fraction?
Some schools teach this clever rule called `Keep Change Flip`:
Keep the numerator the same,
Change the operation from division to multiplication,
Flip the bottom Fraction.

Answer 12

\[\large \lim_{x\to7}\frac{1}{(x-7)^2}\quad\approx\quad 1\cdot\frac{99999^2}{1}\]

Answer 13

yes kcf is the best in complex quotient rule questions, and that would be 9,999,800,001 or just 99999^2 if i shouldve left it

Answer 14

Well the point is,
when you plug in a number really small for x,
when it gets really close to zero,

OVERALL, the expression is getting HUUUUUUUGE, right?

Answer 15

Yes!

Answer 16

Err not really small for x, I mean really close to 7 >.<

Answer 17

oh, yes. That does make sense

Answer 18

So I hope this conclusion makes some sense from that work we did.\[\large\rm \lim_{x\to7}\frac{1}{(x-7)^2}=\infty\]

Answer 19

Yes, it does. Thank you. Now, about the asymptote...

Answer 20

Recall that in the land of math, you can't divide by 0.
\(\large\rm \frac10\) <--- This is not a number.
When see that we're getting close to this type of behavior though,
where the bottom is getting close to zero,
we know that we're near an asymptote.


When you have a rational function,\[\large\rm \frac{x~stuff}{other~x~stuff}\]
We can let the "other x stuff" equal zero to find out where our asymptote is located.

Answer 21

\[\large\rm (x-7)^2=0\]So for the function we were given,
we let our denominator equal zero,
and solve for x to see where this is happening.

Answer 22

7?

Answer 23

\[\large\rm \frac{1}{(x-7)^2}\]So our expression becomes 1/0, this non-existent number, when x=7.

Good, we found the location of our aspymotote!

Answer 24

Yay! I got the question right! Thank you so much. I'm not sure how medals work yet but you explained that very, very well and you didn't talk down to me at all, so if I could give you a thousand medals I would, haha! You made it very clear. It has been a while for me since Algrabra 2 and I feel that was a great refresher so thank you, thank you, thank you!

Answer 25

Haha np.

Medals are no big deal,
it's just another way of saying thanks I guess.
You can give a medal (sort of like a thumbs up) to a person if you feel they've helped you on a question.

\(\large\rm \color{indianred}{\text{Welcome to OpenStudy! :)}}\)

Answer 26

I usually give a medal back as well if the person is responsive.
We get a lot of lazy bums around here :) lolol

Answer 27

Haha, thank you. And I'm glad I joined... I see myself asking many AP calculus questions in the future... lol

Answer 28

Thanks again and have a great night :D