Consider the function f(x)=sin(1/x)
Find a sequence of x-values that approach 0 such that
(1) sin (1/x)=0 {Hint: Use the fact that sin(pi) = sin(2pi)=sin(3pi)=...=sin(npi)=0}
(2) sin (1/x)=1 {Hint: Use the fact that sin(npi)/2)=1 if n= 1,5,9...}
(3) sin (1/x)=-1
(4) Explain why your answers to any of parts(1-3) show that lim X->0 sin(1/x) does not exist.
Is sin sin (1/x)=0 and sin (1/x)=-1 does not exist.
What is sin (1/x)=1 then.
plz answer
To find a sequence of x-values that approach 0 such that sin(1/x) equals a specific value, we can use the given hints and the properties of the sine function. Let's go through each part step by step:
(1) To find x-values that make sin(1/x) equal to 0, we can set 1/x equal to nπ, where n is an integer. Solving for x, we get x = 1/(nπ). Therefore, a sequence of x-values that approach 0 for sin(1/x) = 0 will be x = 1/(nπ), where n = ±1, ±2, ±3, and so on.
(2) To find x-values that make sin(1/x) equal to 1, we can set 1/x equal to (2n + 1)π/2, where n is an integer. Solving for x, we get x = 1/((2n + 1)π/2). Therefore, a sequence of x-values that approach 0 for sin(1/x) = 1 will be x = 1/((2n + 1)π/2), where n = 0, ±1, ±2, ±3, and so on.
(3) To find x-values that make sin(1/x) equal to -1, we can set 1/x equal to nπ, where n is an odd integer. Solving for x, we get x = 1/nπ. Therefore, a sequence of x-values that approach 0 for sin(1/x) = -1 will be x = 1/nπ, where n = ±1, ±3, ±5, and so on.
Now, let's address part (4). The answers to parts (1-3) show that there are infinite possible sequences of x-values that approach 0 for sin(1/x) to reach different values (0, 1, or -1). This indicates that the function sin(1/x) does not have a unique limit as x approaches 0. In other words, the limit of sin(1/x) as x approaches 0 does not exist. The oscillating behavior of sin(1/x) for different sequences of x-values approaching 0 demonstrates this non-existence of the limit.
As for sin(1/x) = 1, there is no x-value that satisfies this equation because the sine function oscillates between -1 and 1, but it never equals 1. Therefore, sin(1/x) = 1 does not have any solutions.