1. ## continuity and limits

consdier the function:

f(x) = (sin3x)/x , x<0
a + 1 , x=0
be^(-2x), x>0

a.)explain why f(x) is continuous for all real x =/ 0 (=/ mean does not equal..sry dunno how to type the sign)

b.)using limits, determine the values of a and b that make f(x) continuous at x = 0

plz help!
thx

2. Originally Posted by ssdimensionss
consdier the function:

f(x) = (sin3x)/x , x<0
a + 1 , x=0
be^(-2x), x>0

a.)explain why f(x) is continuous for all real x =/ 0 (=/ mean does not equal..sry dunno how to type the sign)

b.)using limits, determine the values of a and b that make f(x) continuous at x = 0

plz help!
thx
a.) Use the known properties of each function.

b.) You need to solve:

1. $\lim_{x \rightarrow 0^-} \frac{\sin(3x)}{x} = a + 1 \Rightarrow 3 = a + 1$.

2. $\lim_{x \rightarrow 0^+} b e^{-2x} = a + 1 \Rightarrow b = a + 1$.

I hope calculating the two limits is routine for you.

3. how do u solve the limit x app 0 sin(3x)/x ? coz if i use sandwich rule the limit does not exist?
thx

4. Originally Posted by ssdimensionss
how do u solve the limit x app 0 sin(3x)/x ? coz if i use sandwich rule the limit does not exist?
thx
$\lim_{x \rightarrow 0} \frac{\sin (3x)}{x} = {\color{red}3} \lim_{x \rightarrow 0} \frac{\sin (3x)}{({\color{red}3}x)}$

Substitute t = 3x:

$= 3 \lim_{t \rightarrow 0} \frac{\sin t}{t} = 3 (1) = 3$.

It would be expected that you know $\lim_{t \rightarrow 0} \frac{\sin t}{t} = 1$ and that you can do basic algebraic manipulations of this limit.