We want to determine the limits.
First, we have the limit of 3 sine of the quantity
2x plus pi sine x as x approaches pi over 6.
Since the sine function and the linear function
are continuous
over all real numbers,
we can determine the limit
by performing direct substitution.
This is equal to 3 sine
of the quantity
two times pi over 6
plus pi
times the sine of pi over 6.
And now we begin simplifying.
Two times pi over 6 is 2/6 pi radians
which simplifies to 1/3 pi or pi over 3,
and then we have plus pi times sine of pi over 6 radians
which is equal to 1/2.
We can always check this on the unit circle,
our reference triangle.
Looking at the unit circle,
remember Y is equal to sine theta.
The terminal side of pi over 6 radians
in standard position, intersects the unit circle
at this point, we notice how the Y coordinate is 1/2
sine pi over 6 is equal to 1/2.
This gives us pie times 1/2
And now we simplify inside the parentheses,
we have 1/3 pi
or pi over 3
plus 1/2 pi or pi over 2.
The least common denominator is going to be 6.
We multiply the numerator and denominator of 1/3 by 2
and the numerator and denominator of 1/2 by 3.
Now we have 2/6 pi plus 3/6 pi
which is equal to 5/6 pi.
So now we have 3 sine
5/6 pi.
If we want 5 pi over 6.
Now we need to find the sine of 5/6 pi radians.
Going back to the unit circle.
The terminal side of the angle
in standard position intersects the unit circle
at this point here we'll notice how the Y coordinate
is 1/2 sine 5/6 pi radians equals 1/2
which gives us 3 times 1/2, which is equal to 3/2
if we want 1.5.
Next, we have the limit of five cosine of the quantity
3x plus pi tangent x divided by 2 as x approaches pi over 4.
The cosine and linear function
are continuous over all real numbers
and the tangent function is continuous
at and around pi over 4 radians
and therefore we can determine the limit
by performing direct substitution.
This is equal to 5 times cosine of the quantity 3 times
pi over 4.
If we want 1/4 pi
plus pi times the tangent of pi over 4
divided by 2.
And now let's begin simplifying.
We have five cosine of,
well 3 times pi over 4 is 3/4 pi.
And then we have plus pi times the tangent of pi over 4,
which is equal to 1, all divided by 2.
Let's go ahead and check this tangent function value
on the unit circle.
On the unit circle tangent theta is equal to Y divided by X.
The terminal side of pi over 4 radians
intersects the unit circle at this point on the unit circle.
Notice how X and Y coordinates are equal,
and therefore Y divided by X is equal to 1.
Simplifying inside the parentheses,
we have 3/4 pi
plus 1/2 pi.
The least common denominator is 4.
We multiply the numerator and denominator 1/2 by 2.
This gives us 3/4 pi plus 2/4 pi
but is equal to 5/4 pi.
Now we have 5 times a cosine
of 5/4 pi.
Or if we want 5 pi over 4.
So now we go back to the unit circle
and find the cosine of 5/4 pi radians.
5/4 pi radians intersects the unit circle at this point
on the unit circle.
Cosine theta is equal to X.
Notice X coordinate is negative square root 2 divided
by 2,
giving us 5 times negative square root 2 divided by 2,
which is equal to negative 5 square root 2 divided by 2.
I hope you found this helpful.