Use the graph to estimate the limit:
lim x->0 sin(3x)/x
When x is in degrees
lim x->0 sin(3x)/x = ________
I thought the the answer was (3*180)/pi but it's not... please help... Thanks
Sorry, I am used to working with radians without thinking when dealing with limits and calculus. I have not properly read the instructions (would be fatal in exams!).
To change from degrees to radians, we need to multiply the number of degrees by %pi;/180, so the problem in radians is really:
Lim x->0 sin(3x*π/180)/x
=Lim x-> sin(πx/60)/x
=π/60
To estimate the limit, we can use the fact that as x approaches 0, sin(x) approaches x. We can apply this property to the given expression.
lim x->0 sin(3x)/x
Using the fact mentioned above, we can rewrite sin(3x)/x as 3*sin(3x)/(3x).
Now, let's evaluate this expression as x approaches 0:
lim x->0 3*sin(3x)/(3x)
We can see that as x approaches 0, 3x also approaches 0. Therefore, we can rewrite the expression further as:
lim x->0 3*sin(3x)/(3x) = 3 * lim x->0 (sin(3x))/(3x)
Now, we can directly substitute x = 0 into the expression:
3 * (sin(3 * 0))/(3 * 0) = 3 * (sin(0))/(0) = 3 * (0)/(0)
Here, we encounter an indeterminate form (0/0), which means we cannot directly evaluate the limit using substitution. However, we can still find the limit by applying L'Hôpital's rule.
L'Hôpital's rule states that if the limit of a fraction is an indeterminate form, we can take the derivative of both the numerator and denominator and then evaluate the limit again.
Let's apply L'Hôpital's rule to the expression:
lim x->0 3 * (sin(3x))/(3x) = 3 * lim x->0 (cos(3x) * 3)/(3) [Applying L'Hôpital's rule]
Now, we can directly substitute x = 0 into the expression:
3 * (cos(3 * 0) * 3)/(3) = 3 * (cos(0) * 3)/(3) = 3 * (1 * 3)/(3) = 3
Therefore, the limit of sin(3x)/x as x approaches 0, when x is in degrees, is equal to 3.
Use Lim x->0 sin(x)/x =0
The proof can be done using the sandwich (squeeze) theorem.
Expand
sin(3x)
=3cos²(x) sin(x)-sin³(x)
=sin(x)(3cos²(x)-1+(1-sin²(x)))
=sin(x)(3cos²(x)-1+cos²(x))
=sin(x)(4cos²(x)-1)
As x->0, cos(x)->1, so the limit becomes
Lim x->0 3sin(x)/x
=3 Lim x->0 sin(x)/x
The latter limit can be quoted as a standard limit, or can be proven using the sandwich theorem.