I need help trying to find the left, right, and overall limits for this piecewise function without using a calculator.
f(x) = { sin(x/3) if x≤pi , (x√ 3)/(2pi)if x>pi} when a=pi
well, geez, these are common enough angles . . .
the left limit is sin π/3 = √3/2
the right limit is (π√3)/(2π) = √3/2
Looks like f(x) is continuous
To find the left limit of the function f(x) as x approaches pi, we need to determine the value of the function as x approaches pi from the left side.
Since the function is defined differently for x less than or equal to pi compared to x greater than pi, we need to evaluate each part separately.
First, let's consider the left limit for x approaching pi for the expression sin(x/3).
To find the limit, we substitute pi for x and calculate the value of sin(pi/3):
lim(x→pi-) sin(x/3) = sin(pi/3) = √3/2 (using known trigonometric values)
Now, let's consider the left limit for x approaching pi for the expression (x√3)/(2pi).
To evaluate this limit, we substitute pi for x and calculate the value:
lim(x→pi-) (x√3)/(2pi) = (pi√3)/(2pi) = √3/2
Finally, to find the overall left limit for the piecewise function f(x) as x approaches pi, we compare the values obtained from each part. Since sin(pi/3) = √3/2 is equal to (pi√3)/(2pi) = √3/2, we can conclude that:
lim(x→pi-) f(x) = sin(pi/3) = √3/2
To find the right limit of the function f(x) as x approaches pi, we follow a similar process. Since the function is defined differently for x less than or equal to pi compared to x greater than pi, we evaluate each part separately.
First, let's consider the right limit for x approaching pi for the expression sin(x/3).
To find the limit, we substitute pi for x and calculate the value of sin(pi/3):
lim(x→pi+) sin(x/3) = sin(pi/3) = √3/2 (using known trigonometric values)
Next, let's consider the right limit for x approaching pi for the expression (x√3)/(2pi).
To evaluate this limit, we substitute pi for x and calculate the value:
lim(x→pi+) (x√3)/(2pi) = (pi√3)/(2pi) = √3/2
Once again, we see that the values obtained from each part are equal. Therefore, we can conclude that:
lim(x→pi+) f(x) = sin(pi/3) = √3/2
Finally, to find the overall limit of the piecewise function f(x) as x approaches pi, we compare the left and right limits. Since the left limit and the right limit both equal √3/2, we can conclude that:
lim(x→pi) f(x) = lim(x→pi-) f(x) = lim(x→pi+) f(x) = √3/2