The value of k which makes f(x) = {sin 1/x, x≠0 { k , x=0 continuous at x=0 is? a. 8 b. 1 c. -1 d. none

Step-by-step explanation:

To make the function f(x) = {sin(1/x), x ≠ 0; k, x = 0} continuous at x = 0, we need to find the value of k that ensures the limit of f(x) as x approaches 0 exists and is equal to k.

First, let's find the limit of sin(1/x) as x approaches 0:

lim(x -> 0) sin(1/x)

This limit does not exist because sin(1/x) oscillates wildly as x gets closer to 0. Therefore, in order for the function to be continuous at x = 0, we need to choose k such that it compensates for the oscillations of sin(1/x) as x approaches 0.

A suitable choice for k is 0 because the limit of sin(1/x) as x approaches 0 is undefined, and setting k = 0 ensures that f(x) becomes a continuous function at x = 0.

So, the correct choice is:

d. None (k = 0)

Final answer:

The value of k that would make the function f(x) = sin(1/x) when x ≠0 and f(x) = k when x=0 continuous at x=0 doesn't exist. This is because the limit of sin(1/x) as x approaches 0 is undefined, hence the function cannot be made continuous at x = 0 for any value of k.

Explanation:

To find the value of k that makes the function continuous at x=0, we can apply the definition of continuity, which states that a function, f(x), is continuous at a certain point, x0, if three conditions are met:

• the function is defined at x0
• the limit as x approaches x0 of f(x) exists
• the limit as x approaches x0 of f(x) is equal to f(x0)

In the case of the function f(x) = sin(1/x), the value for x = 0 is undefined, but we've been given that f(0) = k. To make the function continuous at x = 0, the value of k should ideally be equal to the limit of sin(1/x) as x approaches 0.

However, as x approaches 0, sin(1/x) oscillates between -1 and 1, making the limit non-existent. Because the limit does not exist, the function is not continuous at x=0 no matter the chosen value of k. Therefore, the correct answer is (d) None.

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