is the limit as x approaches 0 of sin3x over 3x equal to zero?
sorry--
basically this is my problem:
lim [sin 3x / 4x)
x-> 0
~~~~I multiplied& eventually got to
.75* lim (sin 3x / 3x)
x-> 0
~so i figured since (lim (sinx/x)
x-> 0
was equal to zero, then
lim (sin3x/ 3x) also equaled 0
x-> 0
is that right? thank you !!!
(all of the x-> 0 should be under the "lim" -- just in case the text shifts...)
see below
your preliminary steps are correct
lim sin3x/(4x) as x--> 0
= lim (3/4)(sin3x/(3x))
= 3/4(1)
= 3/4
lim sinx/x = 1 not zero
as x-->0
continued..
Here is a simple way to check your limit answers if you have a calculator
pick a value very "close" to your approach value, in this case I would pick x = .001
evaluate using that value, (you are not yet dividing by zero, but close)
for your question I got .749998875, close to 3/4 I would say.
PS. Make sure your calculator is set to Radians
Yes, you are correct. To determine if the limit of (sin 3x / 3x) as x approaches 0 is equal to 0, we can use the fact that the limit of (sin x / x) as x approaches 0 is equal to 0.
By multiplying both the numerator and denominator by 3, we obtain:
lim [(sin 3x / 3x) * 3 / 3]
x->0
This can be rewritten as:
lim [(sin 3x / x) * 3 / 3]
x->0
Now, notice that the expression in the brackets is the same as the limit we mentioned earlier, which is equal to zero. Therefore, we can substitute the value of that limit into our current limit:
lim [0 * 3 / 3]
x->0
Simplifying this expression gives us:
lim [0]
x->0
And since the limit of a constant value is equal to that constant value, we have:
0
Thus, the limit of (sin 3x / 3x) as x approaches 0 is equal to 0.