Piecewise Function Continuity

Question

theslytherinhelper · Answer

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theslytherinhelper · Answer

I know the answer is -0.47, but I don't understand.
I put in the limit as x approaches -4, but I got -12.47 for the first equation and -12.52 for the second one. Shouldn't this mean that it's not continuous at -0.47?

owlcoffee · Answer

Now, continuity is defined as the lateral limit as x approaches the division point of the divided function:
$$f.cont \iff \lim_{x \rightarrow a ^{\pm}}f(x)=f(a)$$
in other words, for a function to be continuos, both limits have to be equa, these being the lateral limits.
So, for you divided function we will have two limits:
$$\lim_{x \rightarrow (-4)^{+}} 3x +k$$
$$\lim_{x \rightarrow (-4)^-} kx ^{2}-5$$
So, all you have to do is solve each limit (I'll leave that to you), being the solutions:
$$-12+k$$
$$16k-5$$
And by definition, both of these limit results must be equal so we will then make equal those two limits:
$$-12+k=16k-5$$
And now it's just a matter of solving for "k".

theslytherinhelper · Answer

So, would that be best served by me plugging in the values they gave me?

theslytherinhelper · Answer

Because
-12+-0.47=16(-0.47)-5
-12.47=-12.52

And that's where I've been getting stuck :/

owlcoffee · Answer

Well, let's look:
$$-12+k=16k-5$$
$$k-16k=-5+12$$
$$-15k=7$$
$$k=-\frac{ 7 }{ 15 }$$
And that would be approximately 0.47 (if rounded to the nearest decimal).

theslytherinhelper · Answer

OHHHHH wait. That makes so much more sense now. Thank you times a thousand; I can actually see it clearly. I was originally following along with a YouTube video that didn't have a variable besides 'x' and didn't solve algebraically.

Seriously. Can I give you a "lifesaver" point? Because that's been plaguing me all day and now that seems so ridiculously simple.

owlcoffee · Answer

Don't worry about it, I am just here to help ;)
Oh and as friendly tip: Don't use youtube to study math, you can use it to strengthen you understanding, but I would suggest math books, those are written by people who know a lot.

theslytherinhelper · Answer

Thanks a ton; I'll keep that in mind! My course is taught in videos (not on YouTube) and that wasn't helping so I went to the last resort. But no more YouTube if I can avoid it, I promise, haha.

owlcoffee · Answer

Sometimes it's good to do it old-school and grab a book and start reading, it's the best way.
I honestly dislike the new way of teaching math.