Making a piecewise function continuous

[meaganjulie]

Homework Statement

find the values of b and c that make the function f continuous on (-\infty,\infty)

f(x) = \frac{sin2x}{x} if x< 0
3-3c+b(x+1) if 0\leqx<2
5-cx+bx^2 if x\geq 2

Homework Equations

lim as x \rightarrow 0- of \frac{sin2x}{x}
works out to be 0

The Attempt at a Solution

 

[tiny-tim]

Welcome to PF!

Hi meaganjulie! Welcome to PF!

lim as x \rightarrow 0- of \frac{sin2x}{x}
works out to be 0

Nope …

what makes you think that? ​

 

[meaganjulie]

nevermind, that limit is actually 2. the limit for zero from the right also must equal 2, and the right and left limits at x=2 must also match.

thats all I've got.

 

[Char. Limit]

Just figure out what c and b values make the first and second equal at x=0.

The other one follows a similar process.

 

[meaganjulie]

i don't understand how to find the values for c and b because i don't understand how i can use the point x=0 because the first part of the function has no c or b values. i have to use the last to equations at x=2, but how do i know what that limit is?

 

[tiny-tim]

Hi meaganjulie!

(just got up :zzz: …)

i don't understand how to find the values for c and b because i don't understand how i can use the point x=0 because the first part of the function has no c or b values. i have to use the last to equations at x=2, but how do i know what that limit is?

The first equation tells you that the limit at x = 0 must be 2.

The second equation tells you conditions on b and c which agree with that limit (at x = 0), and that gives you a formula (in b and c) for the limit at x = 2.

And the third equation tells you conditions on b and c which agree with that limit at x = 2.