Let f(x) = x*sin(5.8/x), x not equal to Zero and f(x) = L, x = Zero.

Find the value of L so that f(x) is continuous at Zero.

Thanks!!

To find the value of L that makes the function f(x) continuous at x = 0, we need to ensure that the left-hand limit and the right-hand limit of f(x) as x approaches 0 are equal.

First, let's find the left-hand limit (LHL) of f(x) as x approaches 0:

LHL = lim(x->0-) f(x)
= lim(x->0-) x * sin(5.8/x)
= lim(x->0-) (sin(5.8/x) / (1/x)) [by applying L'Hôpital's rule]
= lim(x->0-) (x * cos(5.8/x) * (-5.8/x²)) [by applying L'Hôpital's rule again]
= lim(x->0-) (-5.8 * cos(5.8/x) / x)

Now, let's find the right-hand limit (RHL) of f(x) as x approaches 0:

RHL = lim(x->0+) f(x)
= lim(x->0+) x * sin(5.8/x)
= lim(x->0+) (sin(5.8/x) / (1/x))
= lim(x->0+) (x * cos(5.8/x) * (-5.8/x²))
= lim(x->0+) (-5.8 * cos(5.8/x) / x)

To ensure that the function f(x) is continuous at x = 0, we need the LHL and RHL to be equal. Therefore, we need:

LHL = lim(x->0-) (-5.8 * cos(5.8/x) / x)
= lim(x->0+) (-5.8 * cos(5.8/x) / x)
= RHL

In order for the LHL and RHL to be equal, the limit must exist, so we can evaluate it by substituting x = 0 into the expression.

Let's substitute x = 0:

LHL = lim(x->0-) (-5.8 * cos(5.8/0) / 0) = lim(x->0-) (-5.8 * cos(∞) / 0) = (-5.8 * cos(∞) / 0)
RHL = lim(x->0+) (-5.8 * cos(5.8/0) / 0) = lim(x->0+) (-5.8 * cos(∞) / 0) = (-5.8 * cos(∞) / 0)

Since both LHL and RHL have the same expression, we can see that the value of L must be equal to (-5.8 * cos(∞) / 0), which is undefined.

Therefore, there is no value of L that makes f(x) continuous at x = 0.