Question

Could someone please help explain this to me?
Use the definition of continuity and the properties of limits
to show that the function is continuous at the given number:

Answer 1

\[f(x)= 3x^{4}-5x+\sqrt[3]{x^{2}+4} \]

Answer 2

When a= 2

Answer 3

There is no 'a'. Do you mean x - 2?

Answer 4

Yeah we're assuming a to be the variable (thus, x) I guess

Answer 5

x=2

Answer 6

Well, let's see it? Do you have your definitions and properties?

Answer 7

No. It's just a textbook question, and I wasn't sure how to go about it.

Answer 8

Okay. The limit has to be the same from both sides. Can you demonstrate that?

Answer 9

\[\lim_{x \rightarrow 2^+} 3x^4−5x+\sqrt[3]{x^2+4}\]
\[\lim_{x \rightarrow 2^+} 3(2)^4−5(2)+\sqrt[3]{(2)^2+4}\]
\[\lim_{x \rightarrow 2^+} 40\]

Answer 10

and
\[\lim_{x \rightarrow 2^-} 3x^4−5x+\sqrt[3]{x^2+4}\]
\[\lim_{x \rightarrow 2^-} 3(2)^4−5(2)+\sqrt[3]{(2)^2+4}\]
\[\lim_{x \rightarrow 2^-} 40\]

Answer 11

No good. Substitution is unacceptable until AFTER you demonstrate Continuity. Have you demonstrated that?

Answer 12

Notice also that your two demonstrations are EXACTLY the same. That is not demonstrating how things work from both sides. It was a good try. I'll give you that. :-)

Answer 13

Im unsure as to how to do that

Answer 14

Could you walk me through what to do?

Answer 15

Hi?

Answer 16

Well, if we're going to talk about Continuity, we should start with a MUCH SIMPLER example. Why are we doing this one?

Answer 17

Could you attempt to explain it with this one please?

Answer 18

You didn't answer my question. We won't get along if you don't answer my questions.

The idea is to PROVE that hanging around x = 2 implies f(x) hangs around 40. Normally, this is done be specifying some small deviation from 40 (usually \(\epsilon\)), and demonstrating a fixed relationship that tells you how close you have to be to 2 (usually \(\delta\)). Have you ever seen this done in any example?

Answer 19

Yes, I have. I'm just unsure of how to demonstrate what it's asking. I grasp the concept though.

Answer 20

Also, we're doing this one because it's an assignment due tomorrow and I'm running on 4 hours of sleep over the past 24 hours so I'd like to have assistance with this one, not some other one.

Answer 21

Well, start with \(|f(x) - f(x_{0}| < \epsilon\). Write that out and see what it looks like.

Answer 22

* \(f(x_{0)}\) - not sure where that other parenthesis went.

Answer 23

What is ϵ?

Answer 24

It is an arbitrary small number. I have to question your grasp of the concept if you have to ask that. Coming to terms with this idea while tired is not really a good plan. if \(\epsilon\) is something specific, we will prove nothing. It needs to be arbitrary so that we can claim that it works for everything we might select.

Answer 25

So like infinity

Answer 26

* \(f(x_{0})\) - Ah!! Third time's the charm! Yikes. Too tired to type straight.

Answer 27

Coming to terms with this idea while tired is not really a good plan.

Answer 28

lol

Answer 29

Infinity? Absolutely not. Two thing wrong with this. 1) Infinity is not a number at all. and 2) Reviewing, it appears I did remember to say SMALL!

Answer 30

ok, point taken

Answer 31

So from here, where would we go?

Answer 32

Did you write it out and see what it looks like? These can be a little weird. Each has it';s own ideas.

Answer 33

I did

Answer 34

I think I confused myself moreso

Answer 35

There is an easy part. \(5|x-x_{0}| < 5\delta\)

Answer 36

ok

Answer 37

Then there is a tricky part: \(x^{4} - x_{0}^{4} = (x-x_{0})(x+x_{0})(x^{2} + x_{0}^{2}) = \delta(x+x_{0})(x^{2} + x_{0}^{2})\). Exactly what to do with that is not exactly clear at this point. If this mess were by itself, there would be a little trick, but in the presence of the other stuff. it will take some creativity.

Answer 38

ok

Answer 39

Okay, I'm too tired to do this, tonight. That last "=" should have been "<".

The last part is the real bear. Ripping up that cube root will take the greatest imagination. It's not impossible, we're both just not ready for it tonight.

Good luck. I need to go find my pillow.

Answer 40

Thank you so much!! I really appreciate you time!~~