Determine values of a and b that make the given function continuous .
16sinx÷x
f(x)= a
bcosx
It's a bicewise function
I think we need more information. f(x) = 16 sin x + x for what values of x? f(x) = a, for what values of x? f(x) = b cos x for what values of x?
16sinx/x if x <0
f(x)= a if x=0
bcosx if x>0
a=16
b=16
lim (sinx)/x = 1
therefore lim 16(sinx)/x = 16
so a = 16 and for b cos x to = 16, b = 16
thanks for helping me
To determine the values of a and b that make the given function continuous, we need to ensure that the function is continuous at the points where the two pieces meet, which is when x is equal to 0.
First, let's consider the left piece of the function: 16sinx/x. For this piece to be continuous at x = 0, the limit of the function as x approaches 0 must exist. Evaluating this limit:
lim (x -> 0) 16sinx/x
We can use L'Hôpital's Rule to solve this limit by differentiating the numerator and denominator:
lim (x -> 0) 16cosx/1 = 16cos(0) = 16
So, in order for the left piece to be continuous at x = 0, the value of a should be 16.
Next, let's consider the right piece of the function: bcosx. For this piece to be continuous at x = 0, the value of bcos(0) should be equal to the value of the left piece at x = 0, which we determined to be 16.
bcos(0) = 16
Since cos(0) = 1, we can solve for b:
b * 1 = 16
b = 16
So, the values of a and b that make the given function continuous are a = 16 and b = 16.