Answer:
a. For the function:
f(x) = { sin x/3, if x ≤ π
{ x√3/2π, if x > π
I. To find lim f(x) as x approaches π from the negative side, we need to evaluate f(x) for values of x that are slightly less than π. In this case, since sin(x/3) is a continuous function, we can simply evaluate it at x = π:
lim f(x) as x approaches π- = f(π-) = sin(π/3) = √3/2
II. To find lim f(x) as x approaches π from the positive side, we need to evaluate f(x) for values of x that are slightly greater than π. In this case, we can simply evaluate the other part of the piecewise function at x = π:
lim f(x) as x approaches π+ = f(π+) = π√3/2π = √3/2
III. To find lim f(x) as x approaches π, we need to check whether the left-hand and right-hand limits are equal. In this case, since both the left- and right-hand limits exist and are equal, we have:
lim f(x) as x approaches π = √3/2
b. For the function:
f(x) = (x^2 - 36)/√(x^2 - 12x + 36)
I. To find lim f(x) as x approaches 6 from the negative side, we need to evaluate f(x) for values of x that are slightly less than 6. In this case, we can substitute x = 6 - h, where h is a positive number approaching zero, to get:
lim f(x) as x approaches 6- = lim f(6 - h) as h approaches 0
Substituting x = 6 - h into the function, we get:
f(6 - h) = [(6 - h)^2 - 36]/√[(6 - h)^2 - 12(6 - h) + 36]
= [h^2 - 12h]/√[h^2]
Simplifying the numerator and denominator separately, we get:
f(6 - h) = h(h - 12)/|h|
Since h approaches 0 from the positive side, we have:
lim f(6 - h) as h approaches 0+ = lim h(h - 12)/h as h approaches 0+ = lim (h - 12) as h approaches 0+ = -12
II. To find lim f(x) as x approaches 6 from the positive side, we need to evaluate f(x) for values of x that are slightly greater than 6. In this case, we can substitute x = 6 + h, where h is a positive number approaching zero, to get:
lim f(x) as x approaches 6+ = lim f(6 + h) as h approaches 0
Substituting x = 6 + h into the function, we get:
f(6 + h) = [(6 + h)^2 - 36]/√[(6 + h)^2 - 12(6 + h) + 36]
= [h^2 + 12h]/√[h^2]
Simplifying the numerator and denominator separately, we get:
f(6 + h) = h(h + 12)/|h|
Since h approaches 0 from the positive side, we have:
lim f(6 + h) as h approaches 0+ = lim h(h +
Explanation: