Show complete work Calculators NOT allowed Let fx sin xx Wh

Show complete work. Calculators NOT allowed Let f(x) = sin x/x What is the average rate of change of f on the interval [pi/2, 3 pi/2]? What is lim_x rightarrow D f(x)? Is the line x = 0 a vertical asymptote of f ? Justify your answer using limits.

Solution

f(x)= sinx/x

a) Average rate of change in the interval [pi/2,3pi/2]

sin(3pi/2)/3pi/2 -sin(pi/2)/pi/2/(3pi/2 - pi/2)= -1/3pi/2 -1/pi/2/(3pi/2-pi/2)

                                                                = -1/12150

b) lim_x->infinity(sinx/x)

We have to use squeeze theorem here

And we know that sin x lies between [-1,1]

Therefore -1<=sinx<=1

lim_x->infinity(-1/x)<= lim_x->infinity(sinx/x)<=lim_x->infinity(1/x)

lim_x->infinity(-1/x)=0 and lim_x->infinity(1/x)=0

Therefore lim_x->infinity(sinx/x)=0

c) f(x)= sinx/x

VA is related to the bottom

And to find the VA we have to set bottom to zero

x=0