Question

Find the constant "a", or the constants "a" and "b", such that the function is continuous on the entire real line.

g(x) = {(4sin(x)/x, x<0), (a-2x, x >=0)}

Answer 1

This is killing me... I can't figure out how to do this

Answer 2

well...

Answer 3

\(\large g(x)
\begin{cases}
\cfrac{4sin(x)}{x}&x<0\\
a-2x&x\ge 0
\end{cases}\) right?

Answer 4

Yes

Answer 5

I assume this isn't calculus?

Answer 6

Calc 1

Answer 7

ahhh ok

Answer 8

well.. the thing is

if we know where the function without the variables arrives, when x = 0

then we simply need to make the 2nd one arrive at that same point

if both of them arrive at the same point when x = 0, then we have removed the discontinuity

Answer 9

It's for my AP Calculus BC course... I've looked through examples of similar problems but it's not clicking for me.

Answer 10

so we'd like to know where \(\bf \cfrac{4sin(x)}{x}\) ends up at when x = 0

Answer 11

and ..... I'd say \(\bf g(x)
\begin{cases}
\cfrac{4sin(x)}{x}&x<0\\
a-2x&x\ge 0
\end{cases}
\\ \quad \\
lim_{x\to 0}\quad \cfrac{4sin(x)}{x}\implies lim_{x\to 0}\ 4\cdot lim_{x\to 0}\ \cfrac{sin(x)}{x}\) any ideas?

Answer 12

say that gives you the value of "something"

then we know it arrives at "something" when x = 0

so we then set a - 2x = "something"

and then both of them arrive at that same "something" value and thus the discontinuity would be removed

Answer 13

I'm not sure how to write limits on here, but the limit of 4sin(x)/x as x->0 is equal to 4, correct?

Answer 14

So then a=2x+4?

Answer 15

yeap because \(\bf g(x)
\begin{cases}
\cfrac{4sin(x)}{x}&x<0\\
a-2x&x\ge 0
\end{cases}
\\ \quad \\
lim_{x\to 0}\quad \cfrac{4sin(x)}{x}\implies lim_{x\to 0}\ 4\cdot {\color{brown}{ lim_{x\to 0}\ \cfrac{sin(x)}{x}}}=1\to 4\cdot 1\to 4\)

Answer 16

That's L'Hospital's rule?

Answer 17

so now we know it arrives to 4 when x = 0 so

a - 2x = 4 setting x = 0 then
a - 2(0) = 4

Answer 18

\(\bf {\color{brown}{ lim_{x\to 0}\ \cfrac{sin(x)}{x}}}=1\) <---? nope, is just from the "squeeze theorem"

Answer 19

hmm

Answer 20

https://www.youtube.com/watch?v=VIHupMeoDxA <--- using the squeeze theorem

Answer 21

Oh, yeah I just remembered that proof. So, a=4 would be correct then?