determining values/ continuity

[Guest]

\(\displaystyle \L f(x)\,=\,\left{\begin{array}{lr}\frac{2\sin{(x)}}{x}&\mbox{if}\,x\,<\,0\\ \,&\,\\ a&\mbox{if}\,x\,=\,0\\ \,&\,\\ b\cos{(x)}&\mbox{if}\,x\,>\,0\end{array}\)

Instruction: Determine values of a and b that make the given function continuous.

I guess I need to find the domains to make these three statements true. Statement 1 doesn't even have a or b. Statement 2 can be any number, positive or negative. And the last one b can't be zero because it rearranges into cos x/b.

The back of the book gives "a = b = 2" as the answer. Does this have to do wit the last statement and incorporating cos= pi over 2?

thanks for the ear...

 

[stapel]

I'm afraid I don't follow your reasoning. For instance, "b cos(x)" is not the same as "cos(x/b)", and I don't see where you're getting that either of these is supposed to be set equal to pi/2.

Are you familiar with the definition of "continuous" in the context of functions?

Thank you.

Eliz.

 

[pka]

This is all we need to know!
\(\displaystyle \L
\begin{array}{l}
\lim _{x \to 0{^-}} \frac{{2\sin (x)}}{x} = 2 \\
\lim _{x \to {0^+}} \cos (x) = 1 \\
\end{array}\)

 

[Guest]

continuous means that there is no break in the function..
so all the three statements is the complete continuous function?
i just don't know how to approach this kind of question..


from pka's answer:
so, i pluged into the calc. for the first statement and got 1.99 = 2
and for the third i pluged in cos x and got .998 = 1 so whatever # i plug for b, it's multiplied by 1 so i get the same number as b
so the answer is 2 mainly because of the first statement?

 

[stapel]

continuous means that there is no break in the function....i just don't know how to approach this kind of question.

Evaluate the first bit at the endpoint. Find a value of "a" so that the next bit matches up. Then find a value of "b" so that the last bit matches up at that endpoint.

In other words, find values of "a" and "b" so that the function has "no break in the function".

Eliz.

 

[Guest]

Evaluate the first bit at the endpoint.

evaluate=solve which equals 2

Find a value of "a" so that the next bit matches up.

so i use a=2? i thought a=0.....

Then find a value of "b" so that the last bit matches up at that endpoint.

so i plug in 2 for b which gives 1.99=2

 

[stapel]

For sin(x)/x: What is the limit of sin(x)/x as x tends toward zero? (They should have given you this limit when they were discussing trig limits and derivatives.)

For "a": Why do you think that a is equal to zero...?

For b cos(x): Where is "1.99" coming from? Instead, try evaluating at the endpoint. What is cos(0)?

Eliz.