## gabie1121 2 years ago find the values for c such that the function is continuous at all x. FUNCTION BELOW

1. gabie1121

$f(x) \left\{ \frac{ \sin(cx) }{ x } if x<0 \right\}$

2. gabie1121

$f(x)=(e ^{x}-2c) if x \ge0$

3. gabie1121

those go together. I just couldn't figure out how to get them all in at once

4. zepdrix

In order for this function to be continuous, we need to pick a $$c$$ value that makes the two pieces connect together nicely at $$x=0$$. Imagine railroad tracks, the track needs to be continuous, and can't have any sharp corners or the train will fly off the tracks.

5. gabie1121

Yep, that i get. So do i set both functions equal to 0 and solve for c or something?

6. zepdrix

We want to look at them in the limit. We want to see what is happening when we get closer and closer to 0 from the left side. And also what is happening when we get closer and closer to 0 from the right side. For this function to be continuous, they must be approaching the same value. So we'll set these limits equal to one another.

7. zepdrix

$\large \lim_{x \rightarrow 0^-}f(x) \quad = \quad \lim_{x \rightarrow 0^-} \frac{\sin(cx)}{x}$

8. zepdrix

Understand why our function is sin(cx)/x when we're approaching from the left?

9. zepdrix

The tiny negative (that looks like an exponent) is letting us know we're approaching from the left.

10. gabie1121

yep cuz it says less than 0

11. zepdrix

Ok cool :) so we'll set that equal to the other piece.

12. zepdrix

And solve for c.

13. gabie1121

do we make the other side a limit as well?

14. zepdrix

$\large \lim_{x \rightarrow 0^-}f(x) \qquad \qquad = \qquad \qquad \lim_{x \rightarrow 0^+}f(x)$$\large \lim_{x \rightarrow 0^-} \frac{\sin(cx)}{x} \qquad = \qquad \lim_{x \rightarrow 0^+}e^x-2c$

15. zepdrix

Yes :) the limits need to agree in order for this function to be continuous.

16. gabie1121

alrighty, well the left side is equal to just c, right?

17. zepdrix

Yes very good ^^

18. gabie1121

because if i multiply by C , i can use the rule that says sinx/x=1

19. zepdrix

you remembered your identity i take it hehe

20. gabie1121

yep! the right side is giving me fits though lol

21. zepdrix

Or would it be -1 since we're coming from the left? Hmm I didn't think about that. lemme check real quick.

22. zepdrix

Nah it's still 1, my bad.

23. gabie1121

well would the right just be e^x-2? Or can i not do that cuz i'm thinking of derivative rules?

24. gabie1121

well the derivative is the limit as x approaches 0, so shouldn't e^x, stay e^x?

25. zepdrix

no we're not thinking of this as a derivative :) We're looking at the limit and saying to ourselves, "If I plug x=0 directly into this function, does it cause a problem?" If the answer is no, then we can do just that!

26. gabie1121

oh! well then we're left with $c=2c$ ?

27. zepdrix

Woops! Recall that if we have a 0 in the exponent, what will that change our base to?

28. zepdrix

Not 0 silly! :O

29. gabie1121

oh well i just plugged in e^0 in my calculator and it gave me 1

30. zepdrix

hah XD that's a way to do it i guess! :D

31. zepdrix

yah 1 :3

32. gabie1121

lol so its actuallly c=1-2c?

33. zepdrix

Yah looks good c:

34. zepdrix

Do you by chance have an answer key that we can check this against? This is one of those annoying problems that it's easy to make a mistake on c: lol

35. gabie1121

c=1/3! , and I don't have one yet. I will monday so its not a huge deal

36. gabie1121

It looks logical enough to me. Thanks, AGAIN lol

37. zepdrix

This type of problem becomes a little bit harder when they throw 2 unknown constants at you. Because then you have to also look at the limits of their derivatives. But this was a good problem to get a feel for the concept c:

38. gabie1121

well i just started calc1 this semester, so i haven't learned much yet. Just dipping my toes in the water. I have to take all the way through calculus 3 though, bleh

39. zepdrix

Hmm you'll do quite well, I can tell. You seem quite smart. You're very quick on remember how to do little steps. Calc 2 is a doozy!! Power Series made me want to rip my hair out! :O

40. gabie1121

Yeah, i'm told i'll want to murder myself with Calc 2. Not looking forward to it, but thanks! That makes me feel a little better

41. gabie1121

i might be hunting you down again! lol