# find constants a and b so function is continuous

• Apr 21st 2010, 07:34 AM
sydewayzlocc
find constants a and b so function is continuous
I am having trouble with this problem :

Find constants a and b so that the function is continuous for all x ∈http://upload.wikimedia.org/math/6/9...41811fd226.png, where

f(x)={sin(ax)/bx, x<0
2, x=0
ax+b, x>0
• Apr 21st 2010, 03:01 PM
ione
Quote:

Originally Posted by sydewayzlocc
I am having trouble with this problem :

Find constants a and b so that the function is continuous for all x ∈http://upload.wikimedia.org/math/6/9...41811fd226.png, where

f(x)={sin(ax)/bx, x<0
2, x=0
ax+b, x>0

f(x) is continuous for $\displaystyle x\neq0$

f(x) is continuous at x = 0 if $\displaystyle \lim_{x\to 0}f(x)=f(0)$

We want $\displaystyle \lim_{x\to 0^-}\frac{sin(ax)}{bx}=2$ and $\displaystyle \lim_{x\to 0^+}ax+b=2$

$\displaystyle \lim_{x\to 0^-}\frac{sin(ax)}{bx}=\lim_{x\to 0^-}\frac{a cos(ax)}{b}=\frac{a}{b}$

$\displaystyle \frac{a}{b}=2$ so $\displaystyle a=2b$

$\displaystyle \lim_{x\to 0^+}ax+b=b$

$\displaystyle b=2$ and $\displaystyle a=4$