Question

{ sinx, if x pi/6

1.limf(x)
a approaches pi/6-
2.limf(x)
a approaches pi/6+
3.limf(x)
a approaches pi/6

Answer 1

Explanation:

Given that x = π/6, we know that sin(x) = sin(π/6) = 1/2.

Using this information, we can evaluate the limits as follows:

lim f(x) as x approaches π/6-:

This means we are looking at the limit as x approaches π/6 from the left side (i.e., from values less than π/6). Since sin(x) is a continuous function, we can simply evaluate the function at x = π/6 to get the limit from the left side. Therefore,

lim f(x) as x approaches π/6- = sin(π/6) = 1/2.

lim f(x) as x approaches π/6+:

This means we are looking at the limit as x approaches π/6 from the right side (i.e., from values greater than π/6). Again, since sin(x) is continuous, we can evaluate the function at x = π/6 to get the limit from the right side. Therefore,

lim f(x) as x approaches π/6+ = sin(π/6) = 1/2.

lim f(x) as x approaches π/6:

This is the overall limit as x approaches π/6, and it exists if and only if the limits from the left and right side are equal. Since we found that both limits from the left and right sides are equal to 1/2, we can conclude that:

lim f(x) as x approaches π/6 = 1/2.