Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.lim x→∞ x tan(8/x)

To find the limit of the function as x approaches infinity, we can use L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

The given function is x * tan(8/x). We can rewrite this as a quotient: (x/1) / (cot(8/x)).

Now, we can apply L'Hospital's Rule. We take the derivative of the numerator and the derivative of the denominator.

The derivative of x with respect to x is 1.

The derivative of cot(x) with respect to x is -csc^2(x), and we have to apply the chain rule because we have cot(8/x). So, the derivative of cot(8/x) with respect to x is -csc^2(8/x) * (-8/x^2).

So, the limit of the function as x approaches infinity is equal to the limit as x approaches infinity of (1) / (-csc^2(8/x) * (-8/x^2)).

As x approaches infinity, 8/x approaches 0. Therefore, csc(0) is undefined, so the limit of the function as x approaches infinity is undefined.