Question

Find the indicated limit, if it exists

Answer 1

ill help right now

Answer 2

We have a piecewise function here. In order for the limit to exist, it has to approach the same value from the lefthand side and the righthand side.
The function at the top is the left hand side, because it goes from -inf to 9.
The function at the bottom is the right hand side, goes from 9 to inf.

When you plug 9 in for x, do both parts of piecewise equal the same?

Answer 3

yes

Answer 4

would the limit be 9 then?

Answer 5

No, try again. :3
If you plug in 9 for x into each part of the piecewise function, what would the value be?

(9) + 9 = ?
27 - (9) = ?

If the answer are the same, then the limit as x approaches 9 will be that answer.

Answer 6

so the limit would be the answer? So 18

Answer 7

Yes, the limit as x approaches 9 will be 18.
Since the piecewise function approaches 18 from both the left and right, as proven in the plug-n-chug, then the limit exists. :)

Answer 8

What about the limit of this question? @iPwnBunnies

Answer 9

Hmm, ok. Let's do the simple plug-n-chug, let's see if the limit exists like that.
What happens when you plug 7 in for x?

Answer 10

7-7 = 0

Answer 11

Ok, sorry.
So yeah, when you plug in 7, you get 7-7 in the denominator.
This is sorta like a rule of limits:
\[\lim_{x \rightarrow 7}\frac{1}{(7-7)^{2}} = \lim_{x \rightarrow 7} \frac{1}{0} = infinity \]

Answer 12

There is no other way to simplify the function.
So, if you take the limit of something, and you get 0 in the denominator after trying EVERYTHING, then the limit will approach infinity.

Answer 13

http://www.wolframalpha.com/input/?i=limit+as+x+approaches+7+of+1%2F%28x-7%29%5E2

Answer 14

Limit = infinity
Vertical Asymptope = 7

Answer 15

asymptote*

Answer 16

Yeah, there will also be a vertical asymptote at x = 7

Answer 17

what if we approached the limit from the left side? will the limit still be infinity?

Answer 18

or would it be - infinity?

Answer 19

Take a look at the graph I gave you the link to. It's kinda difficult to do it mathematically.
Do you see how the graph approaches infinity from both the left and the right of the asymptote?

Answer 20

aha. Thanks!