Can anyone help with one question on calc assignment?

Question

F(x)= (sin2x) / x , x cannot equal zero

k, x=0

In order for f(x) to be continuous at x=0, what must the value of k be?

Answer 1

take the left hand limit and the right hand limit of the function as x →0.

limit sin(2x)/x as x→0 is an indeterminate form so use L'Hôpital's Rule

lim 2cos(x)/1 as x→0 and this evaluates to 2 from the right and the left.

so if k = 2 at x = 0 then you'll have a continuous function.

Answer 2

In order for f to be continuous, the value of the limit of the function at every point must equal the value of the function itself at that point. This will be true autoumatically for x not equal to 0 because there f is the composition of two continuous functions. So the question boils down to finding the limit of f(x) as x approaches 0. One way to do it is this:
Let z = 2x
Then limit, x-> 0 of f(x) = limit z->0 f(z)

limit z->0 f(z) = limit z->0 sin(z)/(z/2) = limit z->0 2 sin(z)/z
= 2 * limit z->0 sin(z)/z
The last limit you have probably seen before and is equal to 1, so
Then limit, x-> 0 of f(x) = 2*(1) = 2
so k must equal 2

Answer 3

If x is in the neighbourood of zero, then sin2x is close to 2x, so sin2x/x is close to 2. In fact, if you define F(0), the fundtion F(x) ends up as being continuous and infintely differentiable there.

Answer 4

lim F(x)=2

2=2cos2x

cos 2x=1

2x=0
x=0

Answer 5

lim [as x->0](sin2x)/x = 2 lim [as x->0](sin2x)/2x =
= 2*1 = 2

Take k=2